Multiplication and Division

Excerpts

Multiplication and Division
**To multiply a polynomial by a monomial**, therefore, Theorem Multiply each term of the polynomial by the monomial, and add the partial products.
Multiplication and Division
Arrange both the dividend and divisor in ascending or descending powers of some common letter.
Multiplication and Division
It is of fundamental importance to arrange the dividend and divisor *in the same order* with respect to a common letter, and *to keep this order throughout the operation*.
Multiplication and Division
The pupil should observe that, with a view to bringing like terms of the partial products in columns, the terms of the multiplicand and multiplier are arranged in the *same order*.
Multiplication and Division
The pupil will notice that by this process we have in effect separated the dividend into two parts, $x^{2} + 7x$ and $11x + 77$, and divided each part by $x + 7$, and that the complete quotient is the sum of the partial quotients $x$ and $11$.
Multiplication and Division
The first term of the dividend is $an$; that is, the product of $a$, the first term of the divisor, by $n$, the first term of the quotient.

Equations

Multiplication and Division
a(b + c) &= ab + ac
A monomial times a sum equals the sum of the monomial times each term, for positive numbers and, the book says, for negative numbers too.
This equation is in Introduction (Introduction)
Multiplication and Division
M(m + n + p) = Mm + Mn + Mp
Multiplying a sum of terms m, n, p by a multiplier M gives the sum of the products of M with each term; this is the step used to multiply polynomials.
Multiplication and Division
a(b + c - d) &= ab + ac - ad
A monomial times a polynomial equals the sum of the monomial times each term, with signs kept, which is the basis for dividing a polynomial by a monomial.

Problems

Exercise 21

Exercise 21, problem 1, p. 53
$x + 7$ and $x$.
Printed answer:
- $x^{2} + 7x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 7*x
Exercise 21, problem 10, p. 53
$2 x^{3} - 3x^{2}$ and $2x^{2}$.
Printed answer:
- $4x^{5} - 6x^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**5 - 6*x**4
Exercise 21, problem 11, p. 53
$x^{2} - 3y^{2}$ and $4y$.
Printed answer:
- $4x^{2}y - 12y^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2*y - 12*y**3
Exercise 21, problem 12, p. 53
$x^{2} - 3 y^{2}$ and $-x^{2}$.
Printed answer:
- $-x^{4} + 3x^{2}y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -x**4 + 3*x**2*y**2
Exercise 21, problem 13, p. 53
$b^{3} - a^{2}b^{2}$ and $-a^{3}$.
Printed answer:
- $-a^{3}b^{3} + a^{5}b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -a**3*b**3 + a**5*b**2
Exercise 21, problem 14, p. 53
$-a^{2}b^{2} - a^{3}$ and $-a^{2}$.
Printed answer:
- $a^{4}b^{2} + a^{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4*b**2 + a**5
Exercise 21, problem 15, p. 53
$2x^{3} - 3x^{2} + x$ and $2x^{2}$.
Printed answer:
- $4x^{5} - 6x^{4} + 2x^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**5 - 6*x**4 + 2*x**3
Exercise 21, problem 16, p. 53
$a^{2} - 5ab - b^{2}$ and $5ab$.
Printed answer:
- $5a^{3}b - 25a^{2}b^{2} - 5ab^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*a**3*b - 25*a**2*b**2 - 5*a*b**3
Exercise 21, problem 17, p. 53
$a^{3} + 2a^{2}b + 2ab^{2}$ and $a^{2}$.
Printed answer:
- $a^{5} + 2a^{4}b + 2a^{3}b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**5 + 2*a**4*b + 2*a**3*b**2
Exercise 21, problem 18, p. 53
$a^{3} + 2a^{2}b + 2ab^{2}$ and $b^{3}$.
Printed answer:
- $a^{3}b^{3} + 2a^{2}b^{4} + 2ab^{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**3*b**3 + 2*a**2*b**4 + 2*a*b**5
Exercise 21, problem 19, p. 53
$4x^{2} - 6xy - 9y^{2}$ and $2x$.
Printed answer:
- $8x^{3} - 12x^{2}y - 18xy^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*x**3 - 12*x**2*y - 18*x*y**2
Exercise 21, problem 2, p. 53
$2x - 3y$ and $4x$.
Printed answer:
- $8x^{2} - 12xy$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*x**2 - 12*x*y
Exercise 21, problem 20, p. 53
$-x^{2} - 2xy + y^{2}$ and $-y$.
Printed answer:
- $x^{2}y + 2xy^{2} - y^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2*y + 2*x*y**2 - y**3
Exercise 21, problem 21, p. 53
$-a^{3} - a^{2}b^{2} - b^{3}$ and $-a^{2}$.
Printed answer:
- $a^{5} + a^{4}b^{2} + a^{2}b^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**5 + a**4*b**2 + a**2*b**3
Exercise 21, problem 22, p. 53
$-x^{2} + 2xy - y^{2}$ and $-y^{2}$.
Printed answer:
- $x^{2}y^{2} - 2xy^{3} + y^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2*y**2 - 2*x*y**3 + y**4
Exercise 21, problem 23, p. 53
$3 a^{2}b^{2} - 4 ab^{3} + a^{3}b$ and $5 a^{2}b^{2}$.
Printed answer:
- $15a^{4}b^{4} - 20a^{3}b^{5} + 5a^{5}b^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 15*a**4*b**4 - 20*a**3*b**5 + 5*a**5*b**3
Exercise 21, problem 24, p. 53
$-ax^{2} + 3axy^{2} - ay^{4}$ and $-3ay^{2}$.
Printed answer:
- $3a^{2}x^{2}y^{2} - 9a^{2}xy^{4} + 3a^{2}y^{6}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*a**2*x**2*y**2 - 9*a**2*x*y**4 + 3*a**2*y**6
Exercise 21, problem 25, p. 53
$x^{12} - x^{10}y^{3} - x^{3}y^{10}$ and $x^{3}y^{2}$.
Printed answer:
- $x^{15}y^{2} - x^{13}y^{5} - x^{6}y^{12}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**15*y**2 - x**13*y**5 - x**6*y**12
Exercise 21, problem 26, p. 53
$-2x^{3} + 3x^{2}y^{2} - 2xy^{3}$ and $-2x^{2}y^{3}$.
Printed answer:
- $4x^{5}y^{3} - 6x^{4}y^{5} + 4x^{3}y^{6}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**5*y**3 - 6*x**4*y**5 + 4*x**3*y**6
Exercise 21, problem 27, p. 53
$a^{3}x^{2}y^{5} - a^{2}xy^{4} - ay^{3}$ and $a^{7}x^{3}y^{5}$.
Printed answer:
- $a^{10}x^{5}y^{10} - a^{9}x^{4}y^{9} - a^{8}x^{3}y^{8}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**10*x**5*y**10 - a**9*x**4*y**9 - a**8*x**3*y**8
Exercise 21, problem 28, p. 53
$3a^{2}b^{2} - 2ab^{3} + 5a^{3}b$ and $5a^{2}b^{3}$.
Printed answer:
- $15a^{4}b^{5} - 10a^{3}b^{6} + 25a^{5}b^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 15*a**4*b**5 - 10*a**3*b**6 + 25*a**5*b**4
Exercise 21, problem 3, p. 53
$2x - 3y$ and $7y$.
Printed answer:
- $14xy - 21y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 14*x*y - 21*y**2
Exercise 21, problem 4, p. 53
$x - 2a$ and $2a$.
Printed answer:
- $2ax - 4a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a*x - 4*a**2
Exercise 21, problem 5, p. 53
$-x + 3b$ and $-b$.
Printed answer:
- $bx - 3b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes b*x - 3*b**2
Exercise 21, problem 6, p. 53
$2a^{2} - 3ab$ and $-3a$.
Printed answer:
- $-6a^{3} + 9a^{2}b$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -6*a**3 + 9*a**2*b
Exercise 21, problem 7, p. 53
$2x^{2} + 3xz$ and $5z$.
Printed answer:
- $10x^{2}z + 15xz^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 10*x**2*z + 15*x*z**2
Exercise 21, problem 8, p. 53
$a^{2} - 5ab$ and $5ab$.
Printed answer:
- $5a^{3}b - 25a^{2}b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*a**3*b - 25*a**2*b**2
Exercise 21, problem 9, p. 53
$x^{2} - 3 xy$ and $-y^{2}$.
Printed answer:
- $-x^{2}y^{2} + 3xy^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -x**2*y**2 + 3*x*y**3

Exercise 22

Exercise 22, problem 1, p. 56
$x + 7$ and $x + 6$.
Printed answer:
- $x^{2} + 13x + 42$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 13*x + 42
Exercise 22, problem 10, p. 56
$5x - 3$ and $4x - 7$.
Printed answer:
- $20x^{2} - 47x + 21$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 20*x**2 - 47*x + 21
Exercise 22, problem 11, p. 56
$a - 2b$ and $a + 3b$.
Printed answer:
- $a^{2} + ab - 6b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + a*b - 6*b**2
Exercise 22, problem 12, p. 56
$a - 7b$ and $a - 5b$.
Printed answer:
- $a^{2} - 12ab + 35b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - 12*a*b + 35*b**2
Exercise 22, problem 13, p. 56
$5x - 3y$ and $5x - 3y$.
Printed answer:
- $25x^{2} - 30xy + 9y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 25*x**2 - 30*x*y + 9*y**2
Exercise 22, problem 14, p. 56
$x - b$ and $x - c$.
Printed answer:
- $x^{2} - bx - cx + bc$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - b*x - c*x + b*c
Exercise 22, problem 15, p. 56
$2m - p$ and $4m - 3p$.
Printed answer:
- $8m^{2} - 10mp + 3p^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 8*m**2 - 10*m*p + 3*p**2
Exercise 22, problem 16, p. 56
$a + b + c$ and $a - c$.
Printed answer:
- $a^{2} + ab - bc - c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + a*b - b*c - c**2
Exercise 22, problem 17, p. 56
$a^{2} - ab + b^{2}$ and $a^{2} + b^{2}$.
Printed answer:
- $a^{4} - a^{3}b + 2a^{2}b^{2} - ab^{3} + b^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 - a**3*b + 2*a**2*b**2 - a*b**3 + b**4
Exercise 22, problem 18, p. 56
$x^{3} - 3x^{2} + 7$ and $x^{2} - 3$.
Printed answer:
- $x^{5} - 3x^{4} - x^{3} + 16x^{2} - 21$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem x**5 - 3*x**4 - x**3 + 16*x**2 - 21
Exercise 22, problem 19, p. 56
$a^{2} + ab + b^{2}$ and $a - b$.
Printed answer:
- $a^{3} - b^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**3 - b**3
Exercise 22, problem 2, p. 56
$x - 7$ and $x + 6$.
Printed answer:
- $x^{2} - x - 42$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - x - 42
Exercise 22, problem 20, p. 56
$a^{2} - ab + b^{2}$ and $a + b$.
Printed answer:
- $a^{3} + b^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**3 + b**3
Exercise 22, problem 21, p. 56
$x^{2} + 5x - 10$ and $2x^{2} + 3x - 4$.
Printed answer:
- $2x^{4} + 13x^{3} - 9x^{2} - 50x + 40$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**4 + 13*x**3 - 9*x**2 - 50*x + 40
Exercise 22, problem 22, p. 56
$3x^{3} - 2x^{2} + x$ and $3x^{2} + 2x - 2$.
Printed answer:
- $9x^{5} - 7x^{3} + 6x^{2} - 2x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*x**5 - 7*x**3 + 6*x**2 - 2*x
Exercise 22, problem 23, p. 56
$x^{3} + 2x^{2}y + 3xy^{2}$ and $x^{2} - 2xy + y^{2}$.
Printed answer:
- $x^{5} - 4x^{2}y^{3} + 3xy^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**5 - 4*x**2*y**3 + 3*x*y**4
Exercise 22, problem 24, p. 56
$a^{2} - 3ab - b^{2}$ and $-a^{2} + ab + 2b^{2}$.
Printed answer:
- $-a^{4} + 4a^{3}b - 7ab^{3} - 2b^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -a**4 + 4*a**3*b - 7*a*b**3 - 2*b**4
Exercise 22, problem 25, p. 56
$3a^{2}b^{2} + 2 ab^{3} - 5a^{3}b$ and $5a^{2}b^{2} - ab^{3} - b^{4}$.
Printed answer:
- $-25a^{5}b^{3} + 20a^{4}b^{4} + 12a^{3}b^{5} - 5a^{2}b^{6} - 2ab^{7}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -25*a**5*b**3 + 20*a**4*b**4 + 12*a**3*b**5 - 5*a**2*b**6 - 2*a*b**7
Exercise 22, problem 26, p. 56
$a^{2} - 2ab + b^{2}$ and $a^{2} + 2ab + b^{2}$.
Printed answer:
- $a^{4} - 2a^{2}b^{2} + b^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 - 2*a**2*b**2 + b**4
Exercise 22, problem 27, p. 56
$ab + ac + cd$ and $ab - ac + cd$.
Printed answer:
- $a^{2}b^{2} + 2abcd - a^{2}c^{2} + c^{2}d^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2*b**2 + 2*a*b*c*d - a**2*c**2 + c**2*d**2
Exercise 22, problem 28, p. 56
$3x^{2}y^{2} + xy^{3} - 2x^{3}y$ and $x^{2}y^{2} + xy^{3} - 3y^{4}$.
Printed answer:
- $-2x^{5}y^{3} + x^{4}y^{4} + 10x^{3}y^{5} - 8x^{2}y^{6} - 3xy^{7}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -2*x**5*y**3 + x**4*y**4 + 10*x**3*y**5 - 8*x**2*y**6 - 3*x*y**7
Exercise 22, problem 29, p. 56
$x^{2} + 2xy - y^{2}$ and $x^{2} - 2xy + y^{2}$.
Printed answer:
- $x^{4} - 4x^{2}y^{2} + 4xy^{3} - y^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 - 4*x**2*y**2 + 4*x*y**3 - y**4
Exercise 22, problem 3, p. 56
$x + 7$ and $x - 6$.
Printed answer:
- $x^{2} + x - 42$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + x - 42
Exercise 22, problem 30, p. 56
$3x^{2} + xy - y^{2}$ and $x^{2} - 2xy - 3y^{2}$.
Printed answer:
- $3x^{4} - 5x^{3}y - 12x^{2}y^{2} - xy^{3} + 3y^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x**4 - 5*x**3*y - 12*x**2*y**2 - x*y**3 + 3*y**4
Exercise 22, problem 31, p. 56
$a^{2} - 2ab - b^{2}$ and $b^{2} - 2ab - a^{2}$.
Printed answer:
- $-a^{4} + 6a^{2}b^{2} - b^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -a**4 + 6*a**2*b**2 - b**4
Exercise 22, problem 32, p. 56
$a^{2} + b^{2} + c^{2} - ac$ and $a^{2} - b^{2} - c^{2}$.
Printed answer:
- $a^{4} - a^{3}c + ab^{2}c - b^{4} - 2b^{2}c^{2} + ac^{3} - c^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 - a**3*c + a*b**2*c - b**4 - 2*b**2*c**2 + a*c**3 - c**4
Exercise 22, problem 33, p. 56
$a^{2} + 4abx - 4a^{2}b^{2}x^{2}$ and $a^{2} - 4abx + 4a^{2}b^{2}x^{2}$.
Printed answer:
- $a^{4} - 16a^{2}b^{2}x^{2} + 32a^{3}b^{3}x^{3} - 16a^{4}b^{4}x^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 - 16*a**2*b**2*x**2 + 32*a**3*b**3*x**3 - 16*a**4*b**4*x**4
Exercise 22, problem 34, p. 56
$3 a^{2} - 2abx + b^{2}x^{2}$ and $2a^{2} + 3abx - 2b^{2}x^{2}$.
Printed answer:
- $6a^{4} + 5a^{3}bx - 10a^{2}b^{2}x^{2} + 7ab^{3}x^{3} - 2b^{4}x^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 6*a**4 + 5*a**3*b*x - 10*a**2*b**2*x**2 + 7*a*b**3*x**3 - 2*b**4*x**4
Exercise 22, problem 35, p. 56
$2x^{3}y + 4x^{2}y^{2} - 8xy^{3}$ and $2xy^{3} - 3x^{2}y^{2} + 5x^{3}y$.
Printed answer:
- $10x^{6}y^{2} + 14x^{5}y^{3} - 48x^{4}y^{4} + 32x^{3}y^{5} - 16x^{2}y^{6}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 10*x**6*y**2 + 14*x**5*y**3 - 48*x**4*y**4 + 32*x**3*y**5 - 16*x**2*y**6
Exercise 22, problem 4, p. 56
$x - 7$ and $x - 6$.
Printed answer:
- $x^{2} - 13x + 42$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 13*x + 42
Exercise 22, problem 5, p. 56
$x + 8$ and $x - 5$.
Printed answer:
- $x^{2} + 3x - 40$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 3*x - 40
Exercise 22, problem 6, p. 56
$2x + 3$ and $2x + 3$.
Printed answer:
- $4x^{2} + 12x + 9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2 + 12*x + 9
Exercise 22, problem 7, p. 56
$2x - 3$ and $2x - 3$.
Printed answer:
- $4x^{2} - 12x + 9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2 - 12*x + 9
Exercise 22, problem 8, p. 56
$2x + 3$ and $2x - 3$.
Printed answer:
- $4x^{2} - 9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2 - 9
Exercise 22, problem 9, p. 56
$3x - 2$ and $2 - 3x$.
Printed answer:
- $-9x^{2} + 12x - 4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -9*x**2 + 12*x - 4

Exercise 23

Exercise 23, problem 1, p. 58
$2a^{3} - a^{2}$ by $a$.
Printed answer:
- $2a^{2} - a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**2 - a
Exercise 23, problem 10, p. 58
$-5x^{3} + x^{2}y$ by $-x^{2}$.
Printed answer:
- $5x - y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*x - y
Exercise 23, problem 11, p. 58
$2a^{5}x^{3} - 2a^{4}x^{2}$ by $2a^{4}x^{2}$.
Printed answer:
- $ax - 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*x - 1
Exercise 23, problem 12, p. 58
$-x^{2}y - x^{2}y^{2}$ by $-xy$.
Printed answer:
- $x + xy$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x + x*y
Exercise 23, problem 13, p. 58
$9a - 12b + 6c$ by $-3$.
Printed answer:
- $-3a + 4b - 2c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*a + 4*b - 2*c
Exercise 23, problem 14, p. 58
$a^{3}b^{2} - a^{2}b^{5} - a^{4}b^{2}$ by $a^{2}b$.
Printed answer:
- $ab - b^{4} - a^{2}b$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*b - b**4 - a**2*b
Exercise 23, problem 15, p. 58
$3x^{3} - 6x^{2}y - 9xy^{2}$ by $3x$.
Printed answer:
- $x^{2} - 2xy - 3y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 2*x*y - 3*y**2
Exercise 23, problem 16, p. 58
$x^{2}y^{2} - x^{3}y - xy^{3}$ by $xy$.
Printed answer:
- $xy - x^{2} - y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x*y - x**2 - y**2
Exercise 23, problem 17, p. 58
$a^{3} - a^{2}b - ab^{2}$ by $-a$.
Printed answer:
- $-a^{2} + ab + b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -a**2 + a*b + b**2
Exercise 23, problem 18, p. 58
$a^{2}b - ab + ab^{2}$ by $-ab$.
Printed answer:
- $-a + 1 - b$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -a + 1 - b
Exercise 23, problem 19, p. 58
$xy - x^{2}y^{2} + x^{3}y^{3}$ by $-xy$.
Printed answer:
- $-1 + xy - x^{2}y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -1 + x*y - x**2*y**2
Exercise 23, problem 2, p. 58
$42a^{5} - 6a^{2}$ by $6a$.
Printed answer:
- $7a^{4} - a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*a**4 - a
Exercise 23, problem 20, p. 58
$-x^{6} - 2x^{5} - x^{4}$ by $-x^{4}$.
Printed answer:
- $x^{2} + 2x + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 2*x + 1
Exercise 23, problem 21, p. 58
$a^{2}x - abx - acx$ by $ax$.
Printed answer:
- $a - b - c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - b - c
Exercise 23, problem 22, p. 58
$3x^{5}y^{2} - 3x^{4}y^{3} - 3x^{2}y^{4}$ by $3x^{2}y^{2}$.
Printed answer:
- $x^{3} - x^{2}y - y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 - x**2*y - y**2
Exercise 23, problem 23, p. 58
$a^{2}b^{2} - 2ab - 3ab^{3}$ by $ab$.
Printed answer:
- $ab - 2 - 3b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a*b - 2 - 3*b**2
Exercise 23, problem 24, p. 58
$3a^{3}c^{3} + 3a^{2}c - 3ac^{2}$ by $3ac$.
Printed answer:
- $a^{2}c^{2} + a - c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2*c**2 + a - c
Exercise 23, problem 3, p. 58
$21x^{4} + 3x^{2}$ by $3x^{2}$.
Printed answer:
- $7x^{2} + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*x**2 + 1
Exercise 23, problem 4, p. 58
$35m^{4} - 7p^{2}$ by $7$.
Printed answer:
- $5m^{4} - p^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*m**4 - p**2
Exercise 23, problem 5, p. 58
$27x^{5} - 45x^{4}$ by $9x^{2}$.
Printed answer:
- $3x^{3} - 5x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x**3 - 5*x**2
Exercise 23, problem 6, p. 58
$24x^{6} - 8x^{3}$ by $-8x^{3}$.
Printed answer:
- $-3x^{3} + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*x**3 + 1
Exercise 23, problem 7, p. 58
$34x^{3} - 51x^{2}$ by $17x$.
Printed answer:
- $2x^{2} - 3x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*x**2 - 3*x
Exercise 23, problem 8, p. 58
$5x^{5} - 10x^{3}$ by $-5x^{3}$.
Printed answer:
- $-x^{2} + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -x**2 + 2
Exercise 23, problem 9, p. 58
$-3a^{2} - 6ac$ by $-3a$.
Printed answer:
- $a + 2c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + 2*c

Exercise 24

Exercise 24, problem 1, p. 62
$x^{2} + 15x + 56$ by $x + 7$.
Printed answer:
- $x + 8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x + 8
Exercise 24, problem 10, p. 62
$x^{8} + x^{4} + 1$ by $x^{4} - x^{2} + 1$.
Printed answer:
- $x^{4} + x^{2} + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 + x**2 + 1
Exercise 24, problem 11, p. 62
$1 - a^{3}b^{3}$ by $1 - ab$.
Printed answer:
- $1 + ab + a^{2}b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 + a*b + a**2*b**2
Exercise 24, problem 12, p. 62
$x^{3} - 8x-3$ by $x - 3$.
Printed answer:
- $x^{2} + 3x + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 3*x + 1
Exercise 24, problem 13, p. 62
$a^{2} - 2ab + b^{2} - c^{2}$ by $a - b - c$.
Printed answer:
- $a - b + c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - b + c
Exercise 24, problem 14, p. 62
$a^{2} + 2ab + b^{2} - c^{2}$ by $a + b + c$.
Printed answer:
- $a + b - c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + b - c
Exercise 24, problem 15, p. 62
$x^{2} - y^{2} + 2yz - z^{2}$ by $x - y + z$.
Printed answer:
- $x + y - z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x + y - z
Exercise 24, problem 16, p. 62
$c^{4} + 2c^{2} - c + 2$ by $c^{2} - c + 1$.
Printed answer:
- $c^{2} + c + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes c**2 + c + 2
Exercise 24, problem 17, p. 62
$x^{2} - 4y^{2} - 4yz - z^{2}$ by $x + 2y + z$.
Printed answer:
- $x - 2y - z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x - 2*y - z
Exercise 24, problem 18, p. 62
$x^{3} - 6a^{3} + 11a^{2}x - 6ax^{2}$ by $x^{2} + 6a^{2} - 5ax$.
Printed answer:
- $x - a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x - a
Exercise 24, problem 19, p. 62
$a^{2} - 4b^{2} - 9c^{2} + 12bc$ by $a - 3c + 2b$.
Printed answer:
- $a - 2b + 3c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - 2*b + 3*c
Exercise 24, problem 2, p. 62
$x^{2} - 15x + 56$ by $x - 7$.
Printed answer:
- $x - 8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x - 8
Exercise 24, problem 20, p. 62
$2a^{3} - 8a + a^{4} + 12-7a^{2}$ by $2 + a^{2} - 3a$.
Printed answer:
- $a^{2} + 5a + 6$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + 5*a + 6
Exercise 24, problem 21, p. 62
$q^{4} + 6q^{3} + 4 + 12q + 13q^{2}$ by $3q + 2 + q^{2}$.
Printed answer:
- $q^{2} + 3q + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes q**2 + 3*q + 2
Exercise 24, problem 22, p. 62
$27a^{3} - 8b^{3}$ by $3a - 2b$.
Printed answer:
- $9a^{2} + 6ab + 4b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*a**2 + 6*a*b + 4*b**2
Exercise 24, problem 23, p. 62
$a^{4} + 9a^{2} + 15-11a - 7a^{3}$ is divided by $a - 5$.
Printed answer:
- $-65$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes -65
On the STU-32 (STU, rpn):
```
5 ENTER 7 −
```
```
5 × 9 +
```
```
5 × 11 −
```
```
5 × 15 +
```
Calculator: -65E+0; the book prints -65. Run on the calculator core at firmware 628c96c.
Exercise 24, problem 24, p. 62
$7 - 8c^{2} + 5c^{3} + 8c$ is divided by $5c - 3$.
Printed answer:
- $10$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 10
On the STU-32 (STU, rpn):
```
3 ENTER 5 ÷ ENTER ENTER ENTER 5 ×
```
```
8 −
```
```
×
```
```
8 +
```
```
×
```
```
7 +
```
Calculator: +10000E-3; the book prints 10. Run on the calculator core at firmware 628c96c.
Exercise 24, problem 25, p. 62
$3 + 11a^{3} + 30a^{4} - 82a^{2} - 5a$ is divided by $3a^{2} - 4 + 2a$.
Printed answer:
- $7a - 45$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 7*a - 45
Exercise 24, problem 26, p. 62
$2x^{3} - 16x + 10-39x^{2} + 17x^{4}$ is divided by $2 - 5x^{2} - 4x$.
Printed answer:
- $2x^{4}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*x**4
Exercise 24, problem 3, p. 62
$x^{2} + x-56$ by $x - 7$.
Printed answer:
- $x + 8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x + 8
Exercise 24, problem 4, p. 62
$x^{2} - x-56$ by $x + 7$.
Printed answer:
- $x - 8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x - 8
Exercise 24, problem 5, p. 62
$2a^{2} + 11a + 5$ by $2a + 1$.
Printed answer:
- $a + 5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + 5
Exercise 24, problem 6, p. 62
$6a^{2} - 7a-3$ by $2a - 3$.
Printed answer:
- $3a + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*a + 1
Exercise 24, problem 7, p. 62
$4a^{2} + 23a + 15$ by $4a + 3$.
Printed answer:
- $a + 5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + 5
Exercise 24, problem 8, p. 62
$3a^{2} - 4a-4$ by $2 - a$.
Printed answer:
- $-3a - 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*a - 2
Exercise 24, problem 9, p. 62
$x^{4} + x^{2} + 1$ by $x^{2} + x + 1$.
Printed answer:
- $x^{2} - x + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - x + 1

Exercise 25

Exercise 25, problem 1, p. 63
Add $2a^{2} - 3ac - 3ab$; $2b^{2} + 3ac + a^{2}$; $-a^{2} - 2b^{2} + 3ab$.
Printed answer:
- $2a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*a**2
Exercise 25, problem 10, p. 63
Add $-2a^{4} + 3a^{3}b - 4a^{2}b^{2}$; $2a^{3}b - 3a^{2}b^{2}$; $7a^{2}b^{2} + 2a^{4} - b^{4}$.
Printed answer:
- $5a^{3}b - b^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*a**3*b - b**4
Exercise 25, problem 11, p. 63
From $3x^{3} + 5x - 1$ take the sum of $x - 5 + 5x^{2}$ and $3 + 4x -3x^{2}$.
Printed answer:
- $3x^{3} - 2x^{2} + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x**3 - 2*x**2 + 1
Exercise 25, problem 12, p. 63
The minuend is $9c^{2} + 11c - 5$, and the remainder is $6c^{2} - 13c + 7$. What is the subtrahend?
Printed answer:
- $3c^{2} + 24c - 12$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*c**2 + 24*c - 12
Exercise 25, problem 13, p. 63
Find the remainder when $a^{4} + 6b^{4}$ is divided by $a^{2} + 2ab + 2b^{2}$.
Printed answer:
- $2b^{4}$.
unverified: no computed check settled this one (yet)
How it was checked
- other: not a kind the checker handles 2*b**4
Exercise 25, problem 14, p. 63
Multiply $2 - 5x^{2} - 4x$ by $5 + 2x - 3x^{2}$.
Printed answer:
- $10 - 16x - 39x^{2} + 2x^{3} + 15x^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 10 - 16*x - 39*x**2 + 2*x**3 + 15*x**4
Exercise 25, problem 15, p. 63
Divide $a^{6} + a^{5}x + a^{4}x^{2} - a^{3}x^{3} + x^{6}$ by $a^{2} + ax + x^{2}$.
Printed answer:
- $a^{4} - ax^{3} + x^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 - a*x**3 + x**4
Exercise 25, problem 16, p. 63
$ax^{3} - cx + bx^{2} - bx^{3} + cx^{2} - x$.
Printed answer:
- $(a - b)x^{3} + (b + c)x^{2} - (c + 1)x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a - b)*x**3 + (b + c)*x**2 - (c + 1)*x
Exercise 25, problem 17, p. 63
$ax^{4} - 2x + bx^{4} - cx - ax^{3} + bx^{3}$.
Printed answer:
- $(a + b)x^{4} - (a - b)x^{3} - (c + 2)x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + b)*x**4 - (a - b)*x**3 - (c + 2)*x
Exercise 25, problem 18, p. 63
$x^{3} - bx^{2} - cx + bx - cx^{2} + ax^{3}$.
Printed answer:
- $(a + 1)x^{3} - (b + c)x^{2} + (b - c)x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes (a + 1)*x**3 - (b + c)*x**2 + (b - c)*x
Exercise 25, problem 2, p. 63
Subtract $3a^{4} - 2 a^{3}b + 4 a^{2}b^{2}$ from $4b^{4} - 2 ab^{3} + 4 a^{2}b^{2}$.
Printed answer:
- $-3a^{4} + 2a^{3}b - 2ab^{3} + 4b^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes -3*a**4 + 2*a**3*b - 2*a*b**3 + 4*b**4
Exercise 25, problem 3, p. 63
Simplify $x - y - \{z - x - (y - x + z)\}$.
Printed answer:
- $x$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x
Exercise 25, problem 4, p. 63
Multiply $a^{2} + b^{2} + c^{2} - d^{2}$ by $a^{2} + b^{2} - c^{2} + d^{2}$.
Printed answer:
- $a^{4} + 2a^{2}b^{2} + b^{4} - c^{4} + 2c^{2}d^{2} - d^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 + 2*a**2*b**2 + b**4 - c**4 + 2*c**2*d**2 - d**4
Exercise 25, problem 5, p. 63
Divide $10y^{6} + 2 - 12y^{5}$ by $1 + y^{2} - 2y$.
Printed answer:
- $10y^{4} + 8y^{3} + 6y^{2} + 4y + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 10*y**4 + 8*y**3 + 6*y**2 + 4*y + 2
Exercise 25, problem 6, p. 63
If $a = 1$, $b = 2$, and $c = -3$, find the value of $a^{2} + b^{2} + c^{2} + 2ab + 2ac + 2bc$.
Printed answer:
- $0$.
verified: the printed answer passed a computed check
How it was checked
- evaluate: passes 0
Exercise 25, problem 7, p. 63
Simplify $x - (y - z) - \bigl\{4y + [2y - (z - x)]\bigr\}$.
Printed answer:
- $2z - 7y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 2*z - 7*y
Exercise 25, problem 8, p. 63
Multiply $a^{2} + b^{2} + c^{2} - ab - ac - bc$ by $a + b + c$.
Printed answer:
- $a^{3} - 3abc + b^{3} + c^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**3 - 3*a*b*c + b**3 + c**3
Exercise 25, problem 9, p. 63
Divide $16y^{4} - 21x^{2}y^{2} + 21x^{3}y - 10x^{4}$ by $4y^{2} - 5x^{2} + 3xy$.
Printed answer:
- $4y^{2} - 3xy + 2x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*y**2 - 3*x*y + 2*x**2