FACTORING
Excerpts
FACTORING
When we divide a number into two numbers, which multiplied together will give a product equal to the given number, we have found the factors of that number. This process is called **factoring**.
FACTORING
Sometimes a polynomial in the form given has no factor common to all its terms, but has factors common to several terms. It may be possible then to group the terms in parentheses so that there will be a binomial or trinomial factor common to all the terms of the polynomial in its new form.
FACTORING
Write two factors, of which one is the sum and the other the difference of the square roots of the terms.
FACTORING
This is the reverse of the case under Multiplication given in Art. 14.
FACTORING
That the quotient is the square of the first term of the divisor, minus the product of the first term by the second, plus the square of the second.
Equations
FACTORING
2 a b + 6 a c + 4 a d = 2 a \left(b + 3 c + 2 d \right)Taking out the common factor 2a from each term of 2ab + 6ac + 4ad leaves the factored form 2a(b + 3c + 2d).
FACTORING
2am+2ax+bm+bx=(m+x)(2a+b).The four-term polynomial 2am + 2ax + bm + bx factors as (m + x)(2a + b) after grouping terms.
FACTORING
a^8+a^6-a^5-a^3+a^2+1=(a^2+1)(a^6-a^3+1).The polynomial a^8 + a^6 - a^5 - a^3 + a^2 + 1 factors as (a^2 + 1)(a^6 - a^3 + 1).
FACTORING
a^2-b^2=(a+b)(a-b)The difference of the squares of two numbers equals the sum of the numbers times their difference.
FACTORING
x^2 + 14x + 45 = (x + 9)(x + 5)The trinomial x^2 + 14x + 45 factors as (x + 9)(x + 5), since 9 and 5 have sum 14 and product 45.
FACTORING
x^2 - 6x + 5 = (x - 5)(x - 1)The trinomial x^2 - 6x + 5 factors as (x - 5)(x - 1), since -5 and -1 have sum -6 and product 5.
FACTORING
x^2 + 2x - 3 = (x + 3)(x - 1)The trinomial x^2 + 2x - 3 factors as (x + 3)(x - 1), since 3 and -1 have sum 2 and product -3.
FACTORING
x^2 - 8x - 20 = (x - 10)(x + 2)The trinomial x^2 - 8x - 20 factors as (x - 10)(x + 2), since -10 and 2 have sum -8 and product -20.
Problems
Exercise 31
Exercise 31, problem 1
$5a^2 - 25$.
Printed answer:- $5(a^2-5)$
verified: the printed answer passed a computed check
How it was checked
factor: passes5*(a**2 - 5)
Exercise 31, problem 10
$7x - 7x^3 +14x^4$.
Printed answer:- $7x(1-x^2+2x^3)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes7*x*(1 - x**2 + 2*x**3)
Exercise 31, problem 11
$3x^3 - x^2 + x$.
Printed answer:- $x(3x^2-x+1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passesx*(3*x**2 - x + 1)
Exercise 31, problem 12
$a^3 - a^2 y + ay^2$.
Printed answer:- $a(a^2-ay+y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passesa*(a**2 - a*y + y**2)
Exercise 31, problem 13
$3a(x + y) + 5mb(x + y) - 9d^2 x(x + y)$.
Printed answer:- $(x+y)(3a+5mb-9d^2x)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x + y)*(3*a + 5*m*b - 9*d**2*x)
Exercise 31, problem 14
$4(a - b) - 15 x y (a - b) + (a - b) - 5 a^2 b(a - b)$.
Printed answer:- $5(a-b)(1-3xy-a^2b)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes5*(a - b)*(1 - 3*x*y - a**2*b)
Exercise 31, problem 15
$4x^3 y - 12a x^2 - 8 x y^3$.
Printed answer:- $4xy(x^2-3xy-2y^2)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem4*x*y*(x**2 - 3*x*y - 2*y**2)
Exercise 31, problem 16
$6 a x^3 y^5 - 4 a x^2 y^6 + 2 a x y^7 - 2 a^2 x y^9$.
Printed answer:- $2axy^5(3x^2-2xy+y^2-ay^4)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes2*a*x*y**5*(3*x**2 - 2*x*y + y**2 - a*y**4)
Exercise 31, problem 17
$51 x^5 y - 34 x^4 y^2 + 17 x^2 y^4$.
Printed answer:- $17x^2y(3x^3-2x^2y+y^3)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes17*x**2*y*(3*x**3 - 2*x**2*y + y**3)
Exercise 31, problem 18
$6 a^2 b^2 - 3 a^3 b^3 c - 9 a b^3 c + 3 abc^2$.
Printed answer:- $3ab(2ab-a^2b^2c-3b^2c+c^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes3*a*b*(2*a*b - a**2*b**2*c - 3*b**2*c + c**2)
Exercise 31, problem 19
$3 a x^7 - 24 a x + 9 ax^5 - 3 ax^4 - 9 ax^6$.
Printed answer:- $3ax(x^6-8+3x^4-x^3-3x^5)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes3*a*x*(x**6 - 8 + 3*x**4 - x**3 - 3*x**5)
Exercise 31, problem 2
$16 + 64xy$.
Printed answer:- $16(1+4xy)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes16*(1 + 4*x*y)
Exercise 31, problem 20
$27 a^8 b^2 c^3 - 81 a^7 b^3 c^3 + 81 a^6 b^4 c^3 - 27 a^5 b^5 c^3 - 27 a^5 b^2 c^6$.
Printed answer:- $27a^5b^2c^3(a^3-3a^2b+3ab^2-b^3-c^3)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes27*a**5*b**2*c**3*(a**3 - 3*a**2*b + 3*a*b**2 - b**3 - c**3)
Exercise 31, problem 21
A lady bought a ribbon for $m$ cents, some tape for $d$ cents, and some thread for $c$ cents, for which she paid $x$ cents on account. How much remains due?
Printed answer:- $m+d+c-x$ cts.
verified: the printed answer passed a computed check
How it was checked
solve: passes{R: m + d + c - x}
Exercise 31, problem 22
Julia has the same number of beads in each hand. If she should change two from her left hand to her right hand, the right hand would contain twice as many as the left. How many beads has she?
Printed answer:- $12$ beads.
verified: the printed answer passed a computed check
How it was checked
solve: passes{n: 6, B: 12}
Exercise 31, problem 3
$2a - 2a^2$.
Printed answer:- $2a(1-a)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes2*a*(1 - a)
Exercise 31, problem 4
$15a^2 - 225a^4$.
Printed answer:- $15a^2(1-15a^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes15*a**2*(1 - 15*a**2)
Exercise 31, problem 5
$x^3 - x^2$.
Printed answer:- $x^2(x-1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passesx**2*(x - 1)
Exercise 31, problem 6
$3a^2 + a^5$.
Printed answer:- $a^2(3+a^3)$.
verified: the printed answer passed a computed check
How it was checked
factor: passesa**2*(3 + a**3)
Exercise 31, problem 7
$a^2 - ab^2$.
Printed answer:- $a(a-b^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passesa*(a - b**2)
Exercise 31, problem 8
$a^2 + ab$.
Printed answer:- $a(a+b)$.
verified: the printed answer passed a computed check
How it was checked
factor: passesa*(a + b)
Exercise 31, problem 9
$6a^3 + 2a^4 + 4a^5$.
Printed answer:- $2a^3(3+a+2a^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes2*a**3*(3 + a + 2*a**2)
Exercise 32
Exercise 32, problem 1
$ax+ay+bx+by$.
Printed answer:- $(a+b)(x+y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a+b)*(x+y)
Exercise 32, problem 10
$y^3-y^2+y-1$.
Printed answer:- $(y^2 + 1)(y - 1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(y**2+1)*(y-1)
Exercise 32, problem 11
$x^5+x^4-x^3-x^2+x+1$.
Printed answer:- $(x^4 - x^2 + 1)(x + 1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**4-x**2+1)*(x+1)
Exercise 32, problem 12
$a^2x+abx+ac+aby+b^2y+bc$.
Printed answer:- $(ax + by + c)(a + b)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a*x+b*y+c)*(a+b)
Exercise 32, problem 13
$ax-bx+by+cy-cx-ay$.
Printed answer:- $(a - b - c)(x - y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a-b-c)*(x-y)
Exercise 32, problem 14
$3ax+3ay-2bx-2by$.
Printed answer:- $(3a - 2b)(x + y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(3*a-2*b)*(x+y)
Exercise 32, problem 15
$2ax-3by+cy-2ay+3bx-cx$.
Printed answer:- $(2a + 3b - c)(x - y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(2*a+3*b-c)*(x-y)
Exercise 32, problem 16
$6amx+3amy-6anx-3any$.
Printed answer:- $3a(2x + y)(m - n)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes3*a*(2*x+y)*(m-n)
Exercise 32, problem 17
A man had 250 acres of land and 30 houses. After trading $x$ of his houses for $y$ acres of land, how many has he of each?
Printed answer:- $250 + yA.; $30 - x$ horses.
unverified: no computed check settled this one (yet)
How it was checked
other: the record may be misread
Exercise 32, problem 18
What is the number that will become four times as large by adding 36 to it?
Printed answer:- $12$.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation{x: 12}
On the STU-32 (STU, rpn):
36 ENTER 4 ENTER 1 −
÷
Calculator:
+12E+0; the book prints12. Run on the calculator core at firmware628c96c.Exercise 32, problem 19
The fifth and seventh of a number are together equal to 24. What is the number?
Printed answer:- $70$.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation{x: 70}
On the STU-32 (STU, rpn):
5 1/x 7 1/x +
24 x↔y ÷
Calculator:
+6999999999999999999999999999999999E-32; the book prints70. Run on the calculator core at firmware628c96c.Exercise 32, problem 2
$x^2+ax+bx+ab$.
Printed answer:- $(x+b)(x+a)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x+b)*(x+a)
Exercise 32, problem 3
$ax^2+ay^2-bx^2-by^2$.
Printed answer:- $(a-b)(x^2+y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a-b)*(x**2+y**2)
Exercise 32, problem 4
$x^2-ax+5x-5a$.
Printed answer:- $(x+5)(x-a)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x+5)*(x-a)
Exercise 32, problem 5
$ax-bx+ab-x^2$.
Printed answer:- $(a-x)(x+b)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a-x)*(x+b)
Exercise 32, problem 6
$x^2+mxy-4xy-4my^2$.
Printed answer:- $(x-4y)(x+my)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x-4*y)*(x+m*y)
Exercise 32, problem 7
$2x^4-x^3+4x-2$.
Printed answer:- $(x^3+2)(2x-1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**3+2)*(2*x-1)
Exercise 32, problem 8
$mx-ma-nx+na$.
Printed answer:- $(m-n)(x-a)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(m-n)*(x-a)
Exercise 32, problem 9
$x^3+x^2+x+1$.
Printed answer:- $(x^2+1)(x+1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**2+1)*(x+1)
Exercise 33
Exercise 33, problem 1
$x^2-y^2$.
Printed answer:- $(x+y)(x-y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x + y)*(x - y)
Exercise 33, problem 10
$16m^2-9n^2$.
Printed answer:- $(4m - 3n)(4m + 3n)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(4*m - 3*n)*(4*m + 3*n)
Exercise 33, problem 11
$81x^2y^4-25b^2d^2$.
Printed answer:- $(9xy^2 - 5bd)(9xy^2 - 5bd)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(9*x*y**2 - 5*b*d)*(9*x*y**2 - 5*b*d)
Exercise 33, problem 12
$729m^4c^2x^{10}-10,000y^4$.
Printed answer:- $(27m^2cx^5 + 100y^2)(27m^2cx^5 - 100y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(27*m**2*c*x**5 + 100*y**2)*(27*m**2*c*x**5 - 100*y**2)
Exercise 33, problem 13
$121m^2-64x^2$.
Printed answer:- $(11m + 8x)(11m - 8x)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(11*m + 8*x)*(11*m - 8*x)
Exercise 33, problem 14
$x^4-y^4$.
Printed answer:- $(x^2 + y^2)(x + y)(x - y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**2 + y**2)*(x + y)*(x - y)
Exercise 33, problem 15
$m^8 - a^8$.
Printed answer:- $(m^4 + a^4)(m^2 + a^2)(m + a)(m - a)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(m**4 + a**4)*(m**2 + a**2)*(m + a)*(m - a)
Exercise 33, problem 16
$a^4b^8 - l$.
Printed answer:- $(a^4b^4 + 1)(a^2b^2 + 1)(ab + 1)(ab - 1)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(a**4*b**4 + 1)*(a**2*b**2 + 1)*(a*b + 1)*(a*b - 1)
Exercise 33, problem 17
$x^16 - b^16$.
Printed answer:- $(x^8 + b^8)(x^4 + b^4)(x^2 + b^2)(x + b)(x - b)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**8 + b**8)*(x**4 + b**4)*(x**2 + b**2)*(x + b)*(x - b)
Exercise 33, problem 18
$16a^4 - 1$.
Printed answer:- $(4a^2 + 1)(2a + 1)(2a - 1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(4*a**2 + 1)*(2*a + 1)*(2*a - 1)
Exercise 33, problem 19
$a^4 - ax^2$.
Printed answer:- $a(a + x)(a - x)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problema*(a + x)*(a - x)
Exercise 33, problem 2
$m^2-n^2$.
Printed answer:- $(m + n)(m - n)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(m + n)*(m - n)
Exercise 33, problem 20
$5b^4 - 5a^2b^2$.
Printed answer:- $5b^2(b + a)(b - a)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes5*b**2*(b + a)*(b - a)
Exercise 33, problem 21
$ax^2 - ay^2 - bx^2 + by^2$.
Printed answer:- $(a - b)(x + y)(x - y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a - b)*(x + y)*(x - y)
Exercise 33, problem 22
$5ax - 5a^3x + 5ay - 5a^3y$.
Printed answer:- $5a(a + y)(1 + a)(1 - a)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem5*a*(a + y)*(1 + a)*(1 - a)
Exercise 33, problem 23
$m^2 - y^2 - am + ay$.
Printed answer:- $(m + y - a)(m - y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(m + y - a)*(m - y)
Exercise 33, problem 24
$a^2 - x^2 - a - x$.
Printed answer:- $(a - x + 1)(a + x)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(a - x + 1)*(a + x)
Exercise 33, problem 25
By how much does $x$ exceed $3y$?
Printed answer:- $x - 3y$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handlesx - 3*y
Exercise 33, problem 26
Eleven years ago C was three times as old as D whose age was $x$ years. What is C’s present age ?
Printed answer:- $3x + 11$ yrs.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation3*x + 11
Exercise 33, problem 3
$a^2b^4-c^2d^2$.
Printed answer:- $(ab^2 + cd)(ab^2 - cd)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a*b**2 + c*d)*(a*b**2 - c*d)
Exercise 33, problem 4
$m^2p^4-x^6y^4$.
Printed answer:- $(mp^2 - x^3y^2)(mp^2 + x^3y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(m*p**2 - x**3*y**2)*(m*p**2 + x**3*y**2)
Exercise 33, problem 5
$a^6b^2x^4-m^4c^8y^{10}$.
Printed answer:- $(a^3bx^2 + m^2c^4y^{10})(a^3bx^2 - m^2c^4y^{10})$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(a**3*b*x**2 + m**2*c**4*y**10)*(a**3*b*x**2 - m**2*c**4*y**10)
Exercise 33, problem 6
$x^2y^4z^4-c^2d^2m^4$.
Printed answer:- $(xy^2z^2 + cdm^2)(x^2y^2z^2 - cdm^2)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(x*y**2*z**2 + c*d*m**2)*(x**2*y**2*z**2 - c*d*m**2)
Exercise 33, problem 7
$x^4y^2-a^6y^4$.
Printed answer:- $(x^2y +a^3y^2)(x^2y - a^3y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**2*y + a**3*y**2)*(x**2*y - a**3*y**2)
Exercise 33, problem 8
$g^6c^6-x^6z^8$.
Printed answer:- $(g^3c^3 - x^3z^4)(g^3c^3 + x^3z^4)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(g**3*c**3 - x**3*z**4)*(g**3*c**3 + x**3*z**4)
Exercise 33, problem 9
$4a^2-9x^2$.
Printed answer:- $(2a - 3x)(2a + 3x)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(2*a - 3*x)*(2*a + 3*x)
Exercise 34
Exercise 34, problem 1
$x^5 + y^5$
Printed answer:- $(x + y)(x^2 - xy + y^2)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(x + y)*(x**2 - x*y + y**2)
Exercise 34, problem 10
$27 + a^3b^6$
Printed answer:- $(3 + ab^2)(9 -3ab^2 + a^2b^4)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(3 + a*b**2)*(9 - 3*a*b**2 + a**2*b**4)
Exercise 34, problem 11
$y^3 + 1$
Printed answer:- $(y + 1)(y^2 - y + 1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(y + 1)*(y**2 - y + 1)
Exercise 34, problem 12
$1 + b^3c^3$
Printed answer:- $(1 + bc^3)(1 - bc^3 + b^2c^5)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(1 + b*c**3)*(1 - b*c**3 + b**2*c**5)
Exercise 34, problem 13
$\frac{1}{8}a^6b^3+ c^0$
Printed answer:- $(\frac{1}{2}a^2b + c^3)(\frac{1}{4}a^4b^2 - \frac{1}{2}a^2bc^3 + c^5)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(Rational(1,2)*a**2*b + c**3)*(Rational(1,4)*a**4*b**2 - Rational(1,2)*a**2*b*c**3 + c**5)
Exercise 34, problem 14
$\frac{1}{27}x^3+ 1$
Printed answer:- $(\frac{1}{4}x + 1)(\frac{1}{2}x^2 - \frac{1}{4}x + 1)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(Rational(1,4)*x + 1)*(Rational(1,2)*x**2 - Rational(1,4)*x + 1)
Exercise 34, problem 15
$(m + n)^3+ 8$
Printed answer:- $(m + n + 2)\{(m + n)^2 - 2(m + n) + 4\}$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(m + n + 2)*((m + n)**2 - 2*(m + n) + 4)
Exercise 34, problem 16
$1 + (x - y)^3$
Printed answer:- $(1 - x - y)\{1 - (x - y) + (x - y)^2\}$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(1 - x - y)*(1 - (x - y) + (x - y)**2)
Exercise 34, problem 17
$2a^2x^3y^6 + 2a^8b^9$
Printed answer:- $2a^2(xy^2 + a^2b^3) (x^2y^4 - a^2b^3xy^2 + a^4b^6)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes2*a**2*(x*y**2 + a**2*b**3)*(x**2*y**4 - a**2*b**3*x*y**2 + a**4*b**6)
Exercise 34, problem 18
$mx + my - nx - ny$
Printed answer:- $(m - n)(x + y)$
verified: the printed answer passed a computed check
How it was checked
factor: passes(m - n)*(x + y)
Exercise 34, problem 19
$a^3x + b^3x - a^3y - b^3y$
Printed answer:- $(x - y)(a + b)(a^2 - ab + b^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x - y)*(a + b)*(a**2 - a*b + b**2)
Exercise 34, problem 2
$c^3 + d^3$
Printed answer:- $(c + d)(c^2 - cd + d^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(c + d)*(c**2 - c*d + d**2)
Exercise 34, problem 20
$a^6 - b^6$
Printed answer:- $(a + b)(a^2 - ab + b^2)(a - b)(a^2 + ab + b^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a + b)*(a**2 - a*b + b**2)*(a - b)*(a**2 + a*b + b**2)
Exercise 34, problem 21
$729x^6 - 64y^6$
Printed answer:- $(27x^3 + 4y^3)(27x^3 - 4y^3)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(27*x**3 + 4*y**3)*(27*x**3 - 4*y**3)
Exercise 34, problem 22
$1 - x^8$
Printed answer:- $(1 + x^4)(1 + x^2)(1 + x)(1 - x)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(1 + x**4)*(1 + x**2)*(1 + x)*(1 - x)
Exercise 34, problem 23
A had $d$ dollars, but, after giving $26 to B, he had one-third as many as B. How many has B? How many had B at first?
Printed answer:- $3(d-26)$; $3d - 104$ dols.
verified: the printed answer passed a computed check
How it was checked
solve: passes{"b": "3*(d - 26)", "b0": "3*d - 104"}
Exercise 34, problem 24
How many units in $y$ tens?
Printed answer:- $10y$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles10*y
Exercise 34, problem 25
If the sum of two numbers is $x$, and one of the numbers is 8, what is the other number?
Printed answer:- $x - 8$.
verified: the printed answer passed a computed check
How it was checked
solve: passesx - 8
Exercise 34, problem 26
What does $a^3$ mean? What does $3a$ stand for? What is the product of $x^4$ and $0$?
Printed answer:- (none printed)
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles
Exercise 34, problem 3
$a^3 + b^3c^3$
Printed answer:- $(a + bc)(a^2 - abc + b^2c^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a + b*c)*(a**2 - a*b*c + b**2*c**2)
Exercise 34, problem 4
$a^3x^3 + y^3$
Printed answer:- $(ax + y)(a^2x^2 - axy + y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a*x + y)*(a**2*x**2 - a*x*y + y**2)
Exercise 34, problem 5
$8a^5b^3c^6 + m^6$
Printed answer:- $(2abc^2 + m^2)(4a^2b^2c^4 - 2 abc^2m^2 + m^4)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(2*a*b*c**2 + m**2)*(4*a**2*b**2*c**4 - 2*a*b*c**2*m**2 + m**4)
Exercise 34, problem 6
$x^6y^3 + 216a^3$
Printed answer:- $(x^2y^3 + 6a)(x^4y^6 - 6ax^2y^3 + 36a^2)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(x**2*y**3 + 6*a)*(x**4*y**6 - 6*a*x**2*y**3 + 36*a**2)
Exercise 34, problem 7
$a^6 + b^6$
Printed answer:- $(a^2 + b^2)(a^4 - a^2b^2 + b^4)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a**2 + b**2)*(a**4 - a**2*b**2 + b**4)
Exercise 34, problem 8
$64x^6 + y^6$
Printed answer:- $(4x^2+y^2)(16x^4 - 4x^2y^2 + y^4)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(4*x**2 + y**2)*(16*x**4 - 4*x**2*y**2 + y**4)
Exercise 34, problem 9
$x^3 + 8$
Printed answer:- $(x + 2)(x^2 - 2x + 4)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x + 2)*(x**2 - 2*x + 4)
Exercise 35
Exercise 35, problem 1
$x^3 - a^3$
Printed answer:- $(x - a)(x^2 +ax + a^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x - a)*(x**2 + a*x + a**2)
Exercise 35, problem 10
$27 - 64m^3x^6y^3.$
Printed answer:- $(3 - 4mx^2y)(9+12mx^2y + 16m^2x^4y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(3 - 4*m*x**2*y)*(9 + 12*m*x**2*y + 16*m**2*x**4*y**2)
Exercise 35, problem 11
$a^3-b^6.$
Printed answer:- $(a-b^2)(a^2 + ab^2 + b^4)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a - b**2)*(a**2 + a*b**2 + b**4)
Exercise 35, problem 12
$m^6 - x^3.$
Printed answer:- $(m^2 - x)(m^4 + m^2x + x^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(m**2 - x)*(m**4 + m**2*x + x**2)
Exercise 35, problem 13
$x^3 - 1.$
Printed answer:- $(x-1)(x^2 + x + 1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x - 1)*(x**2 + x + 1)
Exercise 35, problem 14
$1 - y^3.$
Printed answer:- $(1-y)(1 + y + y^2)$
verified: the printed answer passed a computed check
How it was checked
factor: passes(1 - y)*(1 + y + y**2)
Exercise 35, problem 15
$\frac{1}{27}x^3y^6 - b^9.$
Printed answer:- $(\frac{1}{3}xy^2 - b^3)(\frac{1}{9}x^3y^4 + \frac{1}{3}xy^2b^3 + b^6)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(Rational(1,3)*x*y**2 - b**3)*(Rational(1,9)*x**3*y**4 + Rational(1,3)*x*y**2*b**3 + b**6)
Exercise 35, problem 16
$8 - \frac{1}{64}m^6n^3.$
Printed answer:- $(2-\frac{1}{4}m^2n)(4+\frac{1}{2}m^2n + \frac{1}{16}m^4n^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(2 - Rational(1,4)*m**2*n)*(4 + Rational(1,2)*m**2*n + Rational(1,16)*m**4*n**2)
Exercise 35, problem 17
$1 - (a + b)^3.$
Printed answer:- $\{1 - (a + b)\} \{1+ (a+b) + (a+b)^2\}$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(1 - (a + b))*(1 + (a + b) + (a + b)**2)
Exercise 35, problem 18
$x^3y^3 - (x - y^2)^3.$
Printed answer:- $\{xy - (x-y^2)\} \{x^2y^2 + xy(x - y^2) + (x - y^2)^2\}$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x*y - (x - y**2))*(x**2*y**2 + x*y*(x - y**2) + (x - y**2)**2)
Exercise 35, problem 19
$x^7 - xy^3.$
Printed answer:- $x(x^2 - y)(x^4 + x^2y + y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passesx*(x**2 - y)*(x**4 + x**2*y + y**2)
Exercise 35, problem 2
$c^3 - b^3$
Printed answer:- $(c - b)(c^2 + cb + b^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(c - b)*(c**2 + c*b + b**2)
Exercise 35, problem 20
$ab^3 - ac^3 + mb^3 - mc^3.$
Printed answer:- $(a + m)(b - c)(b^2 + bc + c^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a + m)*(b - c)*(b**2 + b*c + c**2)
Exercise 35, problem 21
$x^6 - y^6$ into four factors.
Printed answer:- $(x + y)(x^2 - xy + y^2)(x - y)(x^2 + xy + y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x + y)*(x**2 - x*y + y**2)*(x - y)*(x**2 + x*y + y**2)
Exercise 35, problem 22
$x^6 - x^5 + x^3 - x^2 - x + 1.$
Printed answer:- $(x^5 + x^2 - 1)(x - 1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**5 + x**2 - 1)*(x - 1)
Exercise 35, problem 23
$a^2 - b^2 + a^3 -b^3.$
Printed answer:- $(a - b)(a + b + a^2 + ab + b^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a - b)*(a + b + a**2 + a*b + b**2)
Exercise 35, problem 24
$mx^3 + my^3 - x - y.$
Printed answer:- $(x + y)(mx^2 - mxy + my^2 - 1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x + y)*(m*x**2 - m*x*y + m*y**2 - 1)
Exercise 35, problem 25
How many hours will it take $x$ men to dig 75 bushels of potatoes if each man digs $y$ bushels an hour?
Printed answer:- $\frac{75}{xy}$ hrs.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation{h: 75/(x*y)}
Exercise 35, problem 26
If there are $x$ tens, $y$ units, and $z$ hundreds in a number, what will represent the whole number of units?
Printed answer:- $100x + 10x + y$.
unverified: no computed check settled this one (yet)
How it was checked
solve: the printed answer does not match the problem{N: 100*x + 10*x + y}
Exercise 35, problem 3
$a^3 - x^3y^3.$
Printed answer:- $(a - xy)(a^2 + axy + x^2y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a - x*y)*(a**2 + a*x*y + x**2*y**2)
Exercise 35, problem 4
$m^3n^3 - c^3.$
Printed answer:- $(mn - c)(m^2n^2 + mnc + c^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(m*n - c)*(m**2*n**2 + m*n*c + c**2)
Exercise 35, problem 5
$27m^3 - 8x^3.$
Printed answer:- $(3m - 2x)(9m^2 + 6mx + 4x^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(3*m - 2*x)*(9*m**2 + 6*m*x + 4*x**2)
Exercise 35, problem 6
$8x^3 -64y^3$
Printed answer:- $8(x - 2y)(x^2 + 2xy + 4y^2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes8*(x - 2*y)*(x**2 + 2*x*y + 4*y**2)
Exercise 35, problem 7
$64a^3x^6b^3 - 125m^9c^3y^6.$
Printed answer:- $(4ax^2b - 5m^3cy^2)(16a^2x^4b^2 + 20abcm^3x^2y^2 + 25m^6c^2y^4)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(4*a*x**2*b - 5*m**3*c*y**2)*(16*a**2*x**4*b**2 + 20*a*b*c*m**3*x**2*y**2 + 25*m**6*c**2*y**4)
Exercise 35, problem 8
$27b^3c^6y^3 - 216a^9m^6x^6.$
Printed answer:- $27(bc^2y - 2a^3m^2x^2)(b^2c^4y^2 + 2a^3bc^2m^2x^2y + 4a^2m^4x^4)$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem27*(b*c**2*y - 2*a**3*m**2*x**2)*(b**2*c**4*y**2 + 2*a**3*b*c**2*m**2*x**2*y + 4*a**2*m**4*x**4)
Exercise 35, problem 9
$8x^9y^3 - 125.$
Printed answer:- $(2x^3y - 5)(4x^6y^2 + 10x^3y + 25)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(2*x**3*y - 5)*(4*x**6*y**2 + 10*x**3*y + 25)
Exercise 36
Exercise 36, problem 1
$a^2 + 2ax + x^2$.
Printed answer:- $(a + x)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a + x)**2
Exercise 36, problem 10
$4y^2 - 12y + 9$.
Printed answer:- $(2y - 3)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(2*y - 3)**2
Exercise 36, problem 11
$9x^2 + 24x + 16$.
Printed answer:- $(3x + 4)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(3*x + 4)**2
Exercise 36, problem 12
$81a^4 - 18a^2 + 1$.
Printed answer:- $(3a - 1)^2(3a + 1)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(3*a - 1)**2*(3*a + 1)**2
Exercise 36, problem 13
$9c^2 + 66cd + 121d^2$.
Printed answer:- $\left(3c+11d\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(3*c + 11*d)**2
Exercise 36, problem 14
$4y^2 - 36y + 81$.
Printed answer:- $\left(2y-9\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(2*y - 9)**2
Exercise 36, problem 15
$x^4 - 6x^2y + 9y^2$.
Printed answer:- $\left(x^2-3y\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**2 - 3*y)**2
Exercise 36, problem 16
$9 - 12x^2 + 4x^4$.
Printed answer:- $\left(3-2x^2\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(3 - 2*x**2)**2
Exercise 36, problem 17
$16x^2y^4 - 24a^3xy^2 + 9a^6$.
Printed answer:- $\left(4xy^2-3a^8\right)^2$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(4*x*y**2 - 3*a**8)**2
Exercise 36, problem 18
$25b^2 + 30bc^2y + 9c^4y^2$.
Printed answer:- $\left(5b+3c^2y\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(5*b + 3*c**2*y)**2
Exercise 36, problem 19
$49a^2 + 28ax^2y + 4x^4y^2$.
Printed answer:- $\left(7a+2x^2y\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(7*a + 2*x**2*y)**2
Exercise 36, problem 2
$c^2 - 2cd + d^2$.
Printed answer:- $(c - d)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(c - d)**2
Exercise 36, problem 20
$\frac{1}{9}x^4 - \frac{2}{3}x^2y^2 + 4y^4$.
Printed answer:- $\left(\frac{1}{3}x^2-y^2\right)^2$.
unverified: no computed check settled this one (yet)
How it was checked
factor: the printed answer does not match the problem(Rational(1, 3)*x**2 - y**2)**2
Exercise 36, problem 21
$\frac{1}{4}a^2 + 2ab + 4b^2$.
Printed answer:- $\left(\frac{1}{2}a+2b\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(Rational(1, 2)*a + 2*b)**2
Exercise 36, problem 22
$\frac{4}{9}x^6y^2 - \frac{4}{3}x^3yz^4 + z^8$.
Printed answer:- $\left(\frac{2}{3}x^3y-z^4\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(Rational(2, 3)*x**3*y - z**4)**2
Exercise 36, problem 23
$x^9 - 4x^6y^4 + 4x^3y^8$.
Printed answer:- $x^3\left(x^3-2y^4\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passesx**3*(x**3 - 2*y**4)**2
Exercise 36, problem 24
$18a^2y + 24ay^3 + 8y^5$.
Printed answer:- $2y\left(3a+2y^2\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes2*y*(3*a + 2*y**2)**2
Exercise 36, problem 25
$3a^3x^5 - 30a^2bx^4 + 75ab^2x^3$.
Printed answer:- $3ax^3\left(ax-5b\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes3*a*x**3*(a*x - 5*b)**2
Exercise 36, problem 26
$(x+y)^2 - 2a^2(x+y) + a^4$.
Printed answer:- $\left(x+y-a^2\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x + y - a**2)**2
Exercise 36, problem 27
$(m^2 - n^2)^2 - 2(m^4 - n^4) + (m^2 + n^2)^2$.
Printed answer:- $\left\{(m^2-n^2)-(m^2+n^2)\right\}^2$ or $\left(-2n^2\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(-2*n**2)**2
Exercise 36, problem 28
$a^2 + 2ab + b^2 + 6a + 6b$.
Printed answer:- $(a+b)(a+b+6)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a + b)*(a + b + 6)
Exercise 36, problem 29
$x^2 + y^2 - 3x - 2xy + 3y$.
Printed answer:- $(x-y)(x-y-3)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x - y)*(x - y - 3)
Exercise 36, problem 3
$4a^2 + 4ay + y^2$.
Printed answer:- $(2a + y)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(2*a + y)**2
Exercise 36, problem 30
$x^6 + y^6 - 2x^3y^3$.
Printed answer:- $\left(x-y\right)^2\left(x^2+xy+y^2\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x - y)**2*(x**2 + x*y + y**2)**2
Exercise 36, problem 31
$a^2 + 2xy$.
Printed answer:- $y^2$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handlesy**2
Exercise 36, problem 32
$m^6 + 6m^3y$.
Printed answer:- $9y^2$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles9*y**2
Exercise 36, problem 33
$c^2 + d^2$.
Printed answer:- $\pm 2cd$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles2*c*d
Exercise 36, problem 34
$(x - y)^4 + 2(x - y)^2$.
Printed answer:- $1$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles1
Exercise 36, problem 35
$(c - d)^2 - 6(c -d )$.
Printed answer:- $9$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles9
Exercise 36, problem 36
$9x^4y^2 - 12x^2y^3$.
Printed answer:- $4y^4$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles4*y**4
Exercise 36, problem 37
$30xy^4 + 9x^2y^6$.
Printed answer:- $25y^2$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles25*y**2
Exercise 36, problem 38
$25a^2b^2 + 36b^4c^2$.
Printed answer:- $\pm 60ab^3c$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles60*a*b**3*c
Exercise 36, problem 39
$49a^2b^2c^2 + 25a^2b^4d^2$.
Printed answer:- $\pm 70a^2b^3cd$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles70*a**2*b**3*c*d
Exercise 36, problem 4
$a^2 + 4ab + 4b^2$.
Printed answer:- $(a + 2b)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a + 2*b)**2
Exercise 36, problem 40
$a^4 - 22a^2 + 9$ (two numbers).
Printed answer:- $16a^2$ or $28a^2$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles16*a**2
Exercise 36, problem 41
$64a^2 - 177ay + 121y^2$.
Printed answer:- $ay$ or $353ay$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handlesa*y
Exercise 36, problem 42
$a^4 - 10a^2b^2 + 9b^4$.
Printed answer:- $4a^2b^2$ or $16a^2b^2$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles4*a**2*b**2
Exercise 36, problem 43
$x^4 + x^2 + 1$.
Printed answer:- $x^2$ or $-3x^2$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handlesx**2
Exercise 36, problem 44
$a^2 + a + 1$.
Printed answer:- $a$ or $-3a$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handlesa
Exercise 36, problem 45a
A stream flows at the rate of $a$ miles an hour, and a man can row in still water $b$ miles an hour. How far can the man row up the stream in an hour? In 6 hours? How far down the stream in an hour? In 3 hours?
Printed answer:- $b-a$
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handlesb - a
Exercise 36, problem 45b
A stream flows at the rate of $a$ miles an hour, and a man can row in still water $b$ miles an hour. How far can the man row up the stream in an hour? In 6 hours? How far down the stream in an hour? In 3 hours?
Printed answer:- $6(b-a)$
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles6*(b - a)
Exercise 36, problem 45c
A stream flows at the rate of $a$ miles an hour, and a man can row in still water $b$ miles an hour. How far can the man row up the stream in an hour? In 6 hours? How far down the stream in an hour? In 3 hours?
Printed answer:- $b+a$
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handlesb + a
Exercise 36, problem 45d
A stream flows at the rate of $a$ miles an hour, and a man can row in still water $b$ miles an hour. How far can the man row up the stream in an hour? In 6 hours? How far down the stream in an hour? In 3 hours?
Printed answer:- $3(b+a)$ miles.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles3*(b + a)
Exercise 36, problem 46
A cistern can he filled by two pipes in 3 hours and 5 hours respectively. What part of the cistern will be filled by both pipes running together for one hour?
Printed answer:- $\frac{8}{15}$ of the cistern.
verified: the printed answer passed a computed check
How it was checked
evaluate: passesRational(8, 15)
Exercise 36, problem 47a
Nine years ago Henry was three times as old as Julius. If Henry is $b$ years old, how old was Julius then? How old is Julius now?
Printed answer:- $\frac{b-9}{3}$;
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation(b - 9)/3
Exercise 36, problem 47b
Nine years ago Henry was three times as old as Julius. If Henry is $b$ years old, how old was Julius then? How old is Julius now?
Printed answer:- $\frac{b-9}{3}+9$ years.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation(b - 9)/3 + 9
Exercise 36, problem 5
$x^2 - 6cx + 9c^2$.
Printed answer:- $(x - 3c)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x - 3*c)**2
Exercise 36, problem 6
$16x^2 - 8xy + y^2$.
Printed answer:- $(4x - y)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(4*x - y)**2
Exercise 36, problem 7
$a^2 + 2a + 1$.
Printed answer:- $(a + 1)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a + 1)**2
Exercise 36, problem 8
$9 - 6x + x^2$.
Printed answer:- $(3 - x)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(3 - x)**2
Exercise 36, problem 9
$x^2 - 10x + 25$.
Printed answer:- $(x - 5)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x - 5)**2
Exercise 37
Exercise 37, problem 1
$a^2+3a+2$.
Printed answer:- $(a+2)(a+1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a+2)*(a+1)
Exercise 37, problem 10
$a^2-4a-77$.
Printed answer:- $(a-11)(a+7)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a-11)*(a+7)
Exercise 37, problem 11
$x^2-2x-63$.
Printed answer:- $(x-9)(x+7)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x-9)*(x+7)
Exercise 37, problem 12
$a^2+10a-75$.
Printed answer:- $(a+15)(a-5)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a+15)*(a-5)
Exercise 37, problem 13
$a^2-24a+143$.
Printed answer:- $(a-11)(a-13)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a-11)*(a-13)
Exercise 37, problem 14
$x^6-4x^3-117$.
Printed answer:- $(a^3-13)(a^3+9)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**3-13)*(x**3+9)
Exercise 37, problem 15
$x^6+4x^3-77$.
Printed answer:- $(x^3+11)(x^3-7)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**3+11)*(x**3-7)
Exercise 37, problem 16
$30+11x+x^2$.
Printed answer:- $(6+x)(5+x)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(6+x)*(5+x)
Exercise 37, problem 17
$21+10a+a^2$.
Printed answer:- $(7+a)(3+a)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(7+a)*(3+a)
Exercise 37, problem 18
$35-12x+x^2$.
Printed answer:- $(7-x)(5-x)$, or $(x-7)(x-5)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(7-x)*(5-x)
Exercise 37, problem 19
$36-13x+x^2$.
Printed answer:- $(9-x)(4-x)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(9-x)*(4-x)
Exercise 37, problem 2
$x^2+9x+18$.
Printed answer:- $(x+6)(x+3)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x+6)*(x+3)
Exercise 37, problem 20
$c^2+2cd-3d^2$.
Printed answer:- $(c+3d)(c-d)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(c+3*d)*(c-d)
Exercise 37, problem 21
$a^2+8ax+15x^2$.
Printed answer:- $(a+5x)(a+3x)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a+5*x)*(a+3*x)
Exercise 37, problem 22
$x^2-xy-20y^2$.
Printed answer:- $(x-5y)(x+4y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x-5*y)*(x+4*y)
Exercise 37, problem 23
$x^4+x^2y-12y^2$.
Printed answer:- $(x^2+4y)(x^2-3y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**2+4*y)*(x**2-3*y)
Exercise 37, problem 24
$x^4-14x^2y^2+45y^4$.
Printed answer:- $(x^2-5y^2)(x+3y)(x-3y)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x**2-5*y**2)*(x+3*y)*(x-3*y)
Exercise 37, problem 25
$x^5-23x^4+132x^3$.
Printed answer:- $x^3(x-12)(x-11)$.
verified: the printed answer passed a computed check
How it was checked
factor: passesx**3*(x-12)*(x-11)
Exercise 37, problem 26
$a^6-12a^5+35a^4$.
Printed answer:- $a^4(a-7)(a-5)$.
verified: the printed answer passed a computed check
How it was checked
factor: passesa**4*(a-7)*(a-5)
Exercise 37, problem 27
$3ax^4-39ax^2+108a$.
Printed answer:- $3a(x+3)(x-3)(x+2)(x-2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes3*a*(x+3)*(x-3)*(x+2)*(x-2)
Exercise 37, problem 28
$3a^3-12a^2b+12ab^2$.
Printed answer:- $3a\left(a-2b\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passes3*a*(a-2*b)**2
Exercise 37, problem 29
$c^5d^3-c^2-a^2c^3d^3+a^2$.
Printed answer:- $(c-a)(c+a)(cd-1)(c^2d^2+cd+1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(c-a)*(c+a)*(c*d-1)*(c**2*d**2+c*d+1)
Exercise 37, problem 3
$x^2-5x+6$.
Printed answer:- $(x-3)(x-2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x-3)*(x-2)
Exercise 37, problem 30
$ax^2+bx^2-5bx-5ax+6b+6a$.
Printed answer:- $(a+b)(x-3)(x-2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a+b)*(x-3)*(x-2)
Exercise 37, problem 31
A field can be mowed by two men in $a$ hours and $b$ hours respectively. What part of the field can be mowed by both men working together for one hour?
Printed answer:- $\frac{1}{a}+\frac{1}{b}$.
verified: the printed answer passed a computed check
How it was checked
solve: passes1/a + 1/b
Exercise 37, problem 32
In how many days can two men do as much as $x$ men in 7 days?
Printed answer:- $\frac{7x}{2}$ days.
verified: the printed answer passed a computed check
How it was checked
solve: passes7*x/2
Exercise 37, problem 4
$a^2-7a+10$.
Printed answer:- $(a-5)(a-2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a-5)*(a-2)
Exercise 37, problem 5
$y^2-10y+16$.
Printed answer:- $(y-8)(y-2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(y-8)*(y-2)
Exercise 37, problem 6
$c^2-c-6$.
Printed answer:- $(c-3)(c+2)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(c-3)*(c+2)
Exercise 37, problem 7
$x^2+4x-5$.
Printed answer:- $(x+5)(x-1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x+5)*(x-1)
Exercise 37, problem 8
$x^2+5x-6$.
Printed answer:- $(x+6)(x-1)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x+6)*(x-1)
Exercise 37, problem 9
$y^2+8y-65$.
Printed answer:- $(y+13)(y-5)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(y+13)*(y-5)
Exercise 38
Exercise 38, problem 1
Divide $50a+9a^4+24-67a^2$ by $a+a^2-6$.
Printed answer:- $9a^2 - 9a - 4$.
verified: the printed answer passed a computed check
How it was checked
identity: passes9*a**2 - 9*a - 4
Exercise 38, problem 10
$8c^6-d^9$.
Printed answer:- $\left(2c^2 - d^3\right)\left(4c^4 + 2c^2d^3 + d^6\right)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(2*c**2-d**3)*(4*c**4+2*c**2*d**3+d**6)
Exercise 38, problem 11
$1+64x^3$.
Printed answer:- $\left(1 + 4x\right)\left(1 - 4x + 16x^2\right)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(1+4*x)*(1-4*x+16*x**2)
Exercise 38, problem 12
$a^6-1$.
Printed answer:- $\left(a+1\right)\left(a^2 - a + 1\right) \left(a - 1\right)\left(a^2 + a + 1\right)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a+1)*(a**2-a+1)*(a-1)*(a**2+a+1)
Exercise 38, problem 13
$x^3+2x^2-x-2$.
Printed answer:- $\left(x+2\right)\left(x+1\right)\left(x-1\right)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x+2)*(x+1)*(x-1)
Exercise 38, problem 14
$x^5-y^5$.
Printed answer:- $\left(x-y\right)\left(x^4 + x^3y + x^2y^2 + xy^3 + y^4\right)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x-y)*(x**4+x**3*y+x**2*y**2+x*y**3+y**4)
Exercise 38, problem 15
$27+12x+x^2$.
Printed answer:- $\left(9+x\right)\left(3+x\right)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(9+x)*(3+x)
Exercise 38, problem 16
Find the cube root of $96m-64-40m^3+m^6+6m^5$.
Printed answer:- $m^2 + 2m - 4$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handlesm**2 + 2*m - 4
Exercise 38, problem 17
Simplify $\left(m+2n\right)^2-\left(2m-n\right)^2$.
Printed answer:- $-3m^2 + 8mn + 3n^2$.
verified: the printed answer passed a computed check
How it was checked
identity: passes-3*m**2 + 8*m*n + 3*n**2
Exercise 38, problem 18
Simplify $(5+7x)(5-7x)-\left(3+2x\right)^2+(x-9)(x-4)$.
Printed answer:- $52 - 25x - 52x^2$.
verified: the printed answer passed a computed check
How it was checked
identity: passes52 - 25*x - 52*x**2
Exercise 38, problem 19
Simplify $(y+x)(y^2-xy+x^2)-(y-x)(y^2+yx+x^2)$.
Printed answer:- $2x^3$.
verified: the printed answer passed a computed check
How it was checked
identity: passes2*x**3
Exercise 38, problem 2
Find the square root of 12y^5-14y^3+1-8y^4+4y+9y^6.
Printed answer:- $3y^3 + 2y^2 - 2y - 1$.
unverified: no computed check settled this one (yet)
How it was checked
other: not a kind the checker handles3*y**3 + 2*y**2 - 2*y - 1
Exercise 38, problem 20
Find three consecutive numbers whose sum is 57.
Printed answer:- 18; 19; 20.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation{a: 18, b: 19, c: 20}
Exercise 38, problem 21
What number, being increased by three-fifths of itself, will equal twice itself diminished by 24?
Printed answer:- 60.
verified: the printed answer passed a computed check
How it was checked
solve: passes, with the problem read into an equation{x: 60}
On the STU-32 (STU, rpn):
24 ENTER 5 ×
5 ENTER 3 −
÷
Calculator:
+60E+0; the book prints60. Run on the calculator core at firmware628c96c.Exercise 38, problem 22
Find the value of $(4a + 5b) (3a+4b)-a^2b+b^3c^2-c^3$ when $a = 0$, $b = 1$, and $c = 2$.
Printed answer:- 16.
verified: the printed answer passed a computed check
How it was checked
evaluate: passes16
On the STU-32 (STU, rpn):
4 ENTER 0 × 5 ENTER 1 × +
3 ENTER 0 × 4 ENTER 1 × +
×
0 GOLD x² 1 × −
1 ENTER 3 yˣ 2 GOLD x² × +
2 ENTER 3 yˣ −
Calculator:
+1600000000000000000000000000000000E-32; the book prints16. Run on the calculator core at firmware628c96c.Exercise 38, problem 3
Expand $(x+y)(x-y)(x^2+y^2)$.
Printed answer:- $x^4 - y^4$.
verified: the printed answer passed a computed check
How it was checked
identity: passesx**4 - y**4
Exercise 38, problem 4
What must be subtracted from $a^3-5a^2+27a-1$ to produce $a+1$?
Printed answer:- $a^3 - 5a^2 + 26a - 2$.
verified: the printed answer passed a computed check
How it was checked
identity: passesa**3 - 5*a**2 + 26*a - 2
Exercise 38, problem 5
Expand $\left(x^2y-4z^3a^4\right)^3$.
Printed answer:- $x^6y^3 - 12x^4y^2z^3a^4 + 48x^2yz^8a^8 - 64z^9a^{12}$.
unverified: no computed check settled this one (yet)
How it was checked
identity: the printed answer does not match the problemx**6*y**3 - 12*x**4*y**2*z**3*a**4 + 48*x**2*y*z**8*a**8 - 64*z**9*a**12
Exercise 38, problem 6
$x^2-11x+10$.
Printed answer:- $\left(x-10\right)\left(x-1\right)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(x-10)*(x-1)
Exercise 38, problem 7
$\left(a+b\right)^2-\left(c+d\right)^2$.
Printed answer:- $\left(a + b + c + d\right)\left(a + b - c - d\right)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(a+b+c+d)*(a+b-c-d)
Exercise 38, problem 8
$6x^3+4x^2-9x-6$.
Printed answer:- $\left(2x^2 - 3\right)\left(3x + 2\right)$.
verified: the printed answer passed a computed check
How it was checked
factor: passes(2*x**2-3)*(3*x+2)
Exercise 38, problem 9
$9x^8-66x^6+121x^4$.
Printed answer:- $x^4\left(3x^2 - 11\right)^2$.
verified: the printed answer passed a computed check
How it was checked
factor: passesx**4*(3*x**2-11)**2