Multiplication and Division

Excerpts

Multiplication and Division
The square of the sum of two numbers is the sum of their squares plus twice their product.
Multiplication and Division
The square of the difference of two numbers is the sum of their squares minus twice their product.
Multiplication and Division
The middle term of each result has for a coefficient the *algebraic sum* of the second terms of the binomials.
Multiplication and Division
The intermediate step given above may be omitted, and the products written at once by *inspection*.
Multiplication and Division
(-8) + (-7) = -15, (-8)(-7) = +56.
Multiplication and Division
The difference of the squares of two numbers is divisible by the sum, and by the difference, of the numbers.

Equations

Multiplication and Division
\frac{a^{2} - b^{2}}{a + b} = a - b
The difference of two squares divided by their sum equals their difference.
Multiplication and Division
\frac{a^{2} - b^{2}}{a - b} = a + b
The difference of two squares divided by the difference of the numbers equals their sum.
Multiplication and Division
\frac{a^{3} - b^{3}}{a - b} = a^{2} + ab + b^{2}
The difference of two cubes divided by the difference of the numbers equals the sum of their squares plus their product.
Multiplication and Division
\frac{a^{3} + b^{3}}{a + b} = a^{2} - ab + b^{2}
The sum of two cubes divided by the sum of the numbers equals the sum of their squares minus their product.

Problems

Exercise 26

Exercise 26, problem 1, p. 65
$(m + n)^{2}$.
Printed answer:
- $m^{2} + 2mn + n^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes m**2 + 2*m*n + n**2
Exercise 26, problem 10, p. 65
$(2a - 5c)^{2}$.
Printed answer:
- $4a2 - 20ac + 25c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**2 - 20*a*c + 25*c**2
Exercise 26, problem 11, p. 65
$(x + y)(x - y)$.
Printed answer:
- $x^{2} - y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - y**2
Exercise 26, problem 12, p. 65
$(4a - b)(4a + b)$.
Printed answer:
- $16a^{2} - b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 16*a**2 - b**2
Exercise 26, problem 13, p. 65
$(2b - 3c)(2b + 3c)$.
Printed answer:
- $4b^{2} - 9c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*b**2 - 9*c**2
Exercise 26, problem 14, p. 65
$(x + 5b)(x + 5b)$.
Printed answer:
- $x^{2} + 10bx + 25b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 10*b*x + 25*b**2
Exercise 26, problem 15, p. 65
$(y - 2z)(y - 2z)$.
Printed answer:
- $y^{2} - 4yz + 4z^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 - 4*y*z + 4*z**2
Exercise 26, problem 16, p. 65
$(y + 3z)(y - 3z)$.
Printed answer:
- $y^{2} - 9z^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 - 9*z**2
Exercise 26, problem 17, p. 65
$(2a - 3b)(2a + 3b)$.
Printed answer:
- $4a^{2} - 9b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**2 - 9*b**2
Exercise 26, problem 18, p. 65
$(2a - 3b)(2a - 3b)$.
Printed answer:
- $4a^{2} - 12ab + 9b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**2 - 12*a*b + 9*b**2
Exercise 26, problem 19, p. 65
$(2a + 3b)(2a + 3b)$.
Printed answer:
- $4a^{2} + 12ab + 9b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**2 + 12*a*b + 9*b**2
Exercise 26, problem 2, p. 65
$(c - a)^{2}$.
Printed answer:
- $c^{2} - 2ac + a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - 2*a*c + c**2
Exercise 26, problem 20, p. 65
$(5x + 3a)(5x - 3a)$.
Printed answer:
- $25x^{2} - 9a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 25*x**2 - 9*a**2
Exercise 26, problem 3, p. 65
$(a + 2c)^{2}$.
Printed answer:
- $a^{2} + 4ac + 4c^{2}$
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + 4*a*c + 4*c**2
Exercise 26, problem 4, p. 65
$(3a - 2b)^{2}$.
Printed answer:
- $9a^{2} - 12ab + 4b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*a**2 - 12*a*b + 4*b**2
Exercise 26, problem 5, p. 65
$(2a + 3b)^{2}$.
Printed answer:
- $4a^{2} + 12ab + 9b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**2 + 12*a*b + 9*b**2
Exercise 26, problem 6, p. 65
$(a - 3b)^{2}$.
Printed answer:
- $a^{2} - 6ab + 9b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - 6*a*b + 9*b**2
Exercise 26, problem 7, p. 65
$(2x - y)^{2}$.
Printed answer:
- $4x^{2} - 4xy + y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2 - 4*x*y + y**2
Exercise 26, problem 8, p. 65
$(y - 2x)^{2}$.
Printed answer:
- $y^{2} - 4xy + 4x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes y**2 - 4*x*y + 4*x**2
Exercise 26, problem 9, p. 65
$(a + 5b)^{2}$.
Printed answer:
- $a^{2} + 10ab + 25b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + 10*a*b + 25*b**2

Exercise 27

Exercise 27, problem 1, p. 67
$(x + 7)(x + 4)$.
Printed answer:
- $x^{2} + 11x + 28$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 11*x + 28
Exercise 27, problem 10, p. 67
$(a - 3b)(a - 4b)$.
Printed answer:
- $a^{2} - 7ab + 12b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - 7*a*b + 12*b**2
Exercise 27, problem 11, p. 67
$(a^{2} - c)(a^{2} + 2c)$.
Printed answer:
- $a^{4} + a^{2}c - 2c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 + a**2*c - 2*c**2
Exercise 27, problem 12, p. 67
$(x - 17)(x - 3)$.
Printed answer:
- $x^{2} - 20x + 51$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 20*x + 51
Exercise 27, problem 13, p. 67
$(x + 6y)(x - 5y)$.
Printed answer:
- $x^{2} + xy - 30y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + x*y - 30*y**2
Exercise 27, problem 14, p. 67
$(3 + 2x)(3 - x)$.
Printed answer:
- $9 + 3x - 2x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9 + 3*x - 2*x**2
Exercise 27, problem 15, p. 67
$(5 + 2x)(1 - 2x)$.
Printed answer:
- $5 - 8x - 4x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5 - 8*x - 4*x**2
Exercise 27, problem 16, p. 67
$(a - 2b)(a + 3b)$.
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 27, problem 17, p. 67
$(a^{2}b^{2} - x^{2})(a^{2}b^{2} - 5x^{2})$.
Printed answer:
- $a^{4}b^{4} - 6a^{2}b^{2}x^{2} + 5x^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4*b**4 - 6*a**2*b**2*x**2 + 5*x**4
Exercise 27, problem 18, p. 67
$(a^{3}b - ab^{3})(a^{3}b + 5ab^{3})$.
Printed answer:
- $a^{6}b^{2} + 4a^{4}b^{4} - 5a^{2}b^{6}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**6*b**2 + 4*a**4*b**4 - 5*a**2*b**6
Exercise 27, problem 19, p. 67
$(x^{2}y - xy^{2})(x^{2}y - 3xy^{2})$.
Printed answer:
- $x^{4}y^{2} - 4x^{3}y^{3} + 3x^{2}y^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4*y**2 - 4*x**3*y**3 + 3*x**2*y**4
Exercise 27, problem 2, p. 67
$(x - 3)(x + 7)$.
Printed answer:
- $x^{2} + 4x - 21$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 4*x - 21
Exercise 27, problem 20, p. 67
$(x^{2}y + xy^{2})(x^{2}y + xy^{2})$.
Printed answer:
- $x^{4}y^{2} + 2x^{3}y^{3} + x^{2}y^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4*y**2 + 2*x**3*y**3 + x**2*y**4
Exercise 27, problem 21, p. 67
$(x + a)(x + b)$.
Printed answer:
- $x^{2} + (a + b)x + ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + (a + b)*x + a*b
Exercise 27, problem 22, p. 67
$(x + a)(x - b)$.
Printed answer:
- $x^{2} + (a - b)x - ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + (a - b)*x - a*b
Exercise 27, problem 23, p. 67
$(x - a)(x + b)$.
Printed answer:
- $x^{2} - (a - b)x - ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - (a - b)*x - a*b
Exercise 27, problem 24, p. 67
$(x - a)(x - b)$.
Printed answer:
- $x^{2} - (a + b)x + ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - (a + b)*x + a*b
Exercise 27, problem 25, p. 67
$(x + 2a)(x + 2b)$.
Printed answer:
- $x^{2} + (2a + 2b)x + 4ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + (2*a + 2*b)*x + 4*a*b
Exercise 27, problem 26, p. 67
$(x - 2a)(x + 2b)$.
Printed answer:
- $x^{2} - (2a - 2b)x - 4ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - (2*a - 2*b)*x - 4*a*b
Exercise 27, problem 27, p. 67
$(x + 2a)(x - 2b)$.
Printed answer:
- $x^{2} + (2a - 2b)x - 4ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + (2*a - 2*b)*x - 4*a*b
Exercise 27, problem 28, p. 67
$(x - 2a)(x - 2b)$.
Printed answer:
- $x^{2} - (2a + 2b)x + 4ab$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - (2*a + 2*b)*x + 4*a*b
Exercise 27, problem 29, p. 67
$(x - a)(x + 3a)$.
Printed answer:
- $x^{2} + 2ax - 3a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 2*a*x - 3*a**2
Exercise 27, problem 3, p. 67
$(x - 2)(x - 4)$.
Printed answer:
- $x^{2} - 6x + 8$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 6*x + 8
Exercise 27, problem 30, p. 67
$(x - 2a)(x + 3a)$.
Printed answer:
- $x^{2} + ax - 6a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + a*x - 6*a**2
Exercise 27, problem 4, p. 67
$(x - 6)(x - 10)$.
Printed answer:
- $x^{2} - 16x + 60$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - 16*x + 60
Exercise 27, problem 5, p. 67
$(x + 7)(x - 4)$.
Printed answer:
- $x^{2} + 3x - 28$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 3*x - 28
Exercise 27, problem 6, p. 67
$(x + a)(x - 2a)$.
Printed answer:
- $x^{2} - ax - 2a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 - a*x - 2*a**2
Exercise 27, problem 7, p. 67
$(x + 3a)(x - a)$.
Printed answer:
- $x^{2} + 2ax - 3a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2 + 2*a*x - 3*a**2
Exercise 27, problem 8, p. 67
$(a + 3c)(a + 3c)$.
Printed answer:
- $a^{2} + 6ac + 9c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + 6*a*c + 9*c**2
Exercise 27, problem 9, p. 67
$(a + 2x)(a - 4x)$.
Printed answer:
- $a^{2} - 2ax - 8x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - 2*a*x - 8*x**2

Exercise 28

Exercise 28, problem 1, p. 68
$\dfrac{x^{2} - 4}{x - 2}$.
Printed answer:
- $x + 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x + 2
Exercise 28, problem 10, p. 68
$\dfrac{9b^{2} - 1}{3b + 1}$.
Printed answer:
- $3b - 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*b - 1
Exercise 28, problem 11, p. 68
$\dfrac{16x^{4} - 25a^{2}}{4x^{2} - 5a}$.
Printed answer:
- $4x^{2} + 5a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2 + 5*a
Exercise 28, problem 12, p. 68
$\dfrac{16x^{4} - 25a^{2}}{4x^{2} + 5a}$.
Printed answer:
- $4x^{2} - 5a$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2 - 5*a
Exercise 28, problem 13, p. 68
$\dfrac{9x^{2} - 25y^{2}}{3x - 5y}$.
Printed answer:
- $3x + 5y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*x + 5*y
Exercise 28, problem 14, p. 68
$\dfrac{a^{2}-(b - c)^{2}}{a-(b - c)}$.
Printed answer:
- $a + b - c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + b - c
Exercise 28, problem 15, p. 68
$\dfrac{a^{2}-(b - c)^{2}}{a + (b - c)}$.
Printed answer:
- $a - b + c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - b + c
Exercise 28, problem 16, p. 68
$\dfrac{a^{2}-(2b - c)^{2}}{a-(2b - c)}$.
Printed answer:
- $a + 2b - c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + 2*b - c
Exercise 28, problem 17, p. 68
$\dfrac{(5a - 7b)^{2} - 1}{(5a - 7b) - 1}$.
Printed answer:
- $5a - 7b + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*a - 7*b + 1
Exercise 28, problem 18, p. 68
$\dfrac{(5a - 7b)^{2} - 1}{(5a - 7b) + 1}$.
Printed answer:
- $5a - 7b - 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 5*a - 7*b - 1
Exercise 28, problem 19, p. 68
$\dfrac{z^{2}-(x - y)^{2}}{z-(x - y)}$.
Printed answer:
- $z + x - y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes z + x - y
Exercise 28, problem 2, p. 68
$\dfrac{x^{2} - 4}{x + 2}$.
Printed answer:
- $x - 2$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x - 2
Exercise 28, problem 20, p. 68
$\dfrac{z^{2}-(x - y)^{2}}{z + (x - y)}$.
Printed answer:
- $z - x + y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes z - x + y
Exercise 28, problem 21, p. 68
$\dfrac{a^{2}-(2b - c)^{2}}{a + (2b - c)}$.
Printed answer:
- $a - 2b + c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - 2*b + c
Exercise 28, problem 22, p. 68
$\dfrac{(x + 3y)^{2} - z^{2}}{(x + 3y) - z}$.
Printed answer:
- $x + 3y + z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x + 3*y + z
Exercise 28, problem 23, p. 68
$\dfrac{(x + 3y)^{2} - z^{2}}{x + 3y + z}$.
Printed answer:
- $x + 3y - z$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x + 3*y - z
Exercise 28, problem 24, p. 68
$\dfrac{(a + 2b)^{2} - 4c^{2}}{(a + 2b) - 2c}$.
Printed answer:
- $a + 2b + 2c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + 2*b + 2*c
Exercise 28, problem 25, p. 68
$\dfrac{(a + 2b)^{2} - 4c^{2}}{(a + 2b) + 2c}$.
Printed answer:
- $a + 2b - 2c$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + 2*b - 2*c
Exercise 28, problem 26, p. 68
$\dfrac{1 - (3x - 2y)^{2}}{1 + (3x - 2y)}$.
Printed answer:
- $1 - 3x + 2y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 - 3*x + 2*y
Exercise 28, problem 3, p. 68
$\dfrac{a^{2} - 9}{a - 3}$.
Printed answer:
- $a + 3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a + 3
Exercise 28, problem 4, p. 68
$\dfrac{a^{2} - 9}{a + 3}$.
Printed answer:
- $a - 3$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a - 3
Exercise 28, problem 5, p. 68
$\dfrac{c^{2} - 25}{c - 5}$.
Printed answer:
- $c + 5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes c + 5
Exercise 28, problem 6, p. 68
$\dfrac{c^{2} - 25}{c + 5}$.
Printed answer:
- $c - 5$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes c - 5
Exercise 28, problem 7, p. 68
$\dfrac{49x^{2} - y^{2}}{7x - y}$.
Printed answer:
- $7x + y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*x + y
Exercise 28, problem 8, p. 68
$\dfrac{49x^{2} - y^{2}}{7x + y}$.
Printed answer:
- $7x - y$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 7*x - y
Exercise 28, problem 9, p. 68
$\dfrac{9b^{2} - 1}{3b - 1}$.
Printed answer:
- $3b + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 3*b + 1

Exercise 29

Exercise 29, problem 1, p. 69
$\dfrac{1 - x^{3}}{1 - x}$.
Printed answer:
- $1 + x + x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 + x + x**2
Exercise 29, problem 10, p. 69
$\dfrac{a^{3} - 8b^{3}}{a - 2b}$.
Printed answer:
- $a^{2} + 2ab + 4b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + 2*a*b + 4*b**2
Exercise 29, problem 11, p. 69
$\dfrac{a^{3} - 64}{a - 4}$.
Printed answer:
- $a^{2} + 4a + 16$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 + 4*a + 16
Exercise 29, problem 12, p. 69
$\dfrac{a^{9} - 27}{a^{3} - 3}$.
Printed answer:
- $a^{6} + 3a^{3} + 9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**6 + 3*a**3 + 9
Exercise 29, problem 13, p. 69
$\dfrac{a^{12} - x^{6}y^{6}}{a^{4} - x^{2}y^{2}}$.
Printed answer:
- $a^{8} + a^{4}x^{2}y^{2} + x^{4}y^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**8 + a**4*x**2*y**2 + x**4*y**4
Exercise 29, problem 14, p. 69
$\dfrac{x^{15} - a^{9}b^{9}}{x^{5} - a^{3}b^{3}}$.
Printed answer:
- $x^{10} + x^{5}a^{3}b^{3} + a^{6}b^{6}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**10 + x**5*a**3*b**3 + a**6*b**6
Exercise 29, problem 15, p. 69
$\dfrac{27x^{3}y^{3} - z^{12}}{3xy - z^{4}}$.
Printed answer:
- $9x^{2}y^{2} + 3xyz^{4} + z^{8}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*x**2*y**2 + 3*x*y*z**4 + z**8
Exercise 29, problem 16, p. 69
$\dfrac{x^{3}y^{3}z^{3} - 1}{xyz - 1}$.
Printed answer:
- $x^{2}y^{2}z^{2} + xyz + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2*y**2*z**2 + x*y*z + 1
Exercise 29, problem 17, p. 69
$\dfrac{8a^{3}b^{3}c^{3} - 27}{2abc - 3}$.
Printed answer:
- $4a^{2}b^{2}c^{2} - 6abc + 9$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 4*a**2*b**2*c**2 - 6*a*b*c + 9
Exercise 29, problem 18, p. 69
$\dfrac{1 - 64x^{3}y^{3}z^{3}}{1 - 4xyz}$.
Printed answer:
- $1 + 4xyz + 16x^{2}y^{2}z^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 + 4*x*y*z + 16*x**2*y**2*z**2
Exercise 29, problem 2, p. 69
$\dfrac{1 - 8a^{3}}{1 - 2a}$.
Printed answer:
- $1 + 2a + 4a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 + 2*a + 4*a**2
Exercise 29, problem 3, p. 69
$\dfrac{1 - 27c^{3}}{1 - 2c}$.
Printed answer:
- $1 + 3c + 9c^{2}$.
unverified: no computed check settled this one (yet)
How it was checked
- identity: the printed answer does not match the problem 1 + 3*c + 9*c**2
Exercise 29, problem 4, p. 69
$\dfrac{8a^{3} - b^{3}}{2a - b}$.
Printed answer:
- $4a^{2} + 2ab + b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**2 + 2*a*b + b**2
Exercise 29, problem 5, p. 69
$\dfrac{64b^{3} - 27c^{3}}{4b - 3c}$.
Printed answer:
- $16b^{2} + 12bc + 9c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 16*b**2 + 12*b*c + 9*c**2
Exercise 29, problem 6, p. 69
$\dfrac{27x^{3} - 8y^{3}}{3x - 2y}$.
Printed answer:
- $9x^{2} + 6xy + 4y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*x**2 + 6*x*y + 4*y**2
Exercise 29, problem 7, p. 69
$\dfrac{x^{3}y^{3} - z^{3}}{xy - z}$.
Printed answer:
- $x^{2}y^{2} + xyz + z^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2*y**2 + x*y*z + z**2
Exercise 29, problem 8, p. 69
$\dfrac{a^{3}b^{3} - 8}{ab - 2}$.
Printed answer:
- $a^{2}b^{2} + 2ab + 4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2*b**2 + 2*a*b + 4
Exercise 29, problem 9, p. 69
$\dfrac{125a^{3} - b^{3}}{5a - b}$.
Printed answer:
- $25a^{2} + 5ab + b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 25*a**2 + 5*a*b + b**2

Exercise 30

Exercise 30, problem 1, p. 70
$\dfrac{1 + x^{3}}{1 + x}$.
Printed answer:
- $1 - x + x^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 - x + x**2
Exercise 30, problem 10, p. 70
$\dfrac{125a^{3} + b^{3}}{5a + b}$.
Printed answer:
- $25a^{2} - 5ab + b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 25*a**2 - 5*a*b + b**2
Exercise 30, problem 11, p. 70
$\dfrac{a^{3} + 8b^{3}}{a + 2b}$.
Printed answer:
- $a^{2} - 2ab + 4b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2 - 2*a*b + 4*b**2
Exercise 30, problem 12, p. 70
$\dfrac{a^{6} + 64}{a^{2} + 4}$.
Printed answer:
- $a^{4} - 4a^{2} + 16$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**4 - 4*a**2 + 16
Exercise 30, problem 13, p. 70
$\dfrac{a^{9} + 27}{a^{3} + 3}$.
Printed answer:
- $a^{6} - 3a^{3} + 9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**6 - 3*a**3 + 9
Exercise 30, problem 14, p. 70
$\dfrac{8a^{6} + b^{3}}{2a^{2} + b}$.
Printed answer:
- $4a^{4} - 2a^{2}b + b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**4 - 2*a**2*b + b**2
Exercise 30, problem 15, p. 70
$\dfrac{a^{12} + x^{6}y^{6}}{a^{4} + x^{2}y^{2}}$.
Printed answer:
- $a^{8} - a^{4}x^{2}y^{2} + x^{4}y^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**8 - a**4*x**2*y**2 + x**4*y**4
Exercise 30, problem 16, p. 70
$\dfrac{x^{15} + a^{9}b^{9}}{x^{5} + a^{3}b^{3}}$.
Printed answer:
- $x^{10} - x^{5}a^{3}b^{3} + a^{6}b^{6}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**10 - x**5*a**3*b**3 + a**6*b**6
Exercise 30, problem 17, p. 70
$\dfrac{27x^{3}y^{3} + z^{12}}{3xy + z^{4}}$.
Printed answer:
- $9x^{2}y^{2} - 3xyz^{4} + z^{8}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*x**2*y**2 - 3*x*y*z**4 + z**8
Exercise 30, problem 18, p. 70
$\dfrac{x^{3}y^{3}z^{3} + 1}{xyz + 1}$.
Printed answer:
- $x^{2}y^{2}z^{2} - xyz + 1$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2*y**2*z**2 - x*y*z + 1
Exercise 30, problem 19, p. 70
$\dfrac{8a^{3}b^{3}c^{3} + 27}{2abc + 3}$.
Printed answer:
- $4a^{2}b^{2}c^{2} - 6abc + 9$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**2*b**2*c**2 - 6*a*b*c + 9
Exercise 30, problem 2, p. 70
$\dfrac{1 + 8a^{3}}{1 + 2a}$.
Printed answer:
- $1 - 2a + 4a^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 - 2*a + 4*a**2
Exercise 30, problem 20, p. 70
$\dfrac{1 + 64x^{3}y^{3}z^{3}}{1 + 4xyz}$.
Printed answer:
- $1 - 4xyz + 16x^{2}y^{2}z^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 - 4*x*y*z + 16*x**2*y**2*z**2
Exercise 30, problem 21, p. 70
$\dfrac{1 + 27a^{6}b^{3}c^{3}}{1 + 3a^{2}bc}$.
Printed answer:
- $1 - 3a^{2}bc + 9a^{4}b^{2}c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 - 3*a**2*b*c + 9*a**4*b**2*c**2
Exercise 30, problem 22, p. 70
$\dfrac{x^{4} - y^{4}}{x - y}$.
Printed answer:
- $x^{3} + x^{2}y + xy^{2} + y^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 + x**2*y + x*y**2 + y**3
Exercise 30, problem 23, p. 70
$\dfrac{x^{4} - y^{4}}{x + y}$.
Printed answer:
- $x^{3} - x^{2}y + xy^{2} - y^{3}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**3 - x**2*y + x*y**2 - y**3
Exercise 30, problem 24, p. 70
$\dfrac{x^{5} - y^{5}}{x - y}$.
Printed answer:
- $x^{4} + x^{3}y + x^{2}y^{2} + xy^{3} + y^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 + x**3*y + x**2*y**2 + x*y**3 + y**4
Exercise 30, problem 25, p. 70
$\dfrac{x^{5} + y^{5}}{x + y}$.
Printed answer:
- $x^{4} - x^{3}y + x^{2}y^{2} - xy^{3} + y^{4}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**4 - x**3*y + x**2*y**2 - x*y**3 + y**4
Exercise 30, problem 26, p. 70
$\dfrac{x^{6} - y^{6}}{x - y}$.
Printed answer:
- $x^{5} + x^{4}y + x^{3}y^{2} + x^{2}y^{3} + xy^{4} + y^{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**5 + x**4*y + x**3*y**2 + x**2*y**3 + x*y**4 + y**5
Exercise 30, problem 27, p. 70
$\dfrac{x^{6} - y^{6}}{x + y}$.
Printed answer:
- $x^{5} - x^{4}y + x^{3}y^{2} - x^{2}y^{3} + xy^{4} - y^{5}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**5 - x**4*y + x**3*y**2 - x**2*y**3 + x*y**4 - y**5
Exercise 30, problem 3, p. 70
$\dfrac{1 + 27c^{3}}{1 + 3c}$.
Printed answer:
- $1 - 3c + 9c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 1 - 3*c + 9*c**2
Exercise 30, problem 4, p. 70
$\dfrac{8a^{3} + b^{3}}{2a + b}$.
Printed answer:
- $4a^{2} - 2ab + b^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*a**2 - 2*a*b + b**2
Exercise 30, problem 5, p. 70
$\dfrac{64b^{3} + 27c^{3}}{4b + 3c}$.
Printed answer:
- $16b^{2} - 12bc + 9c^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 16*b**2 - 12*b*c + 9*c**2
Exercise 30, problem 6, p. 70
$\dfrac{27x^{3} + 8y^{3}}{3x + 2y}$.
Printed answer:
- $9x^{2} - 6xy + 4y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 9*x**2 - 6*x*y + 4*y**2
Exercise 30, problem 7, p. 70
$\dfrac{8x^{3} + 125y^{3}}{2x + 5y}$.
Printed answer:
- $4x^{2} - 10xy + 25y^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes 4*x**2 - 10*x*y + 25*y**2
Exercise 30, problem 8, p. 70
$\dfrac{x^{3}y^{3} + z^{3}}{xy + z}$.
Printed answer:
- $x^{2}y^{2} - xyz + z^{2}$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes x**2*y**2 - x*y*z + z**2
Exercise 30, problem 9, p. 70
$\dfrac{a^{3}b^{3} + 8}{ab + 2}$.
Printed answer:
- $a^{2}b^{2} - 2ab + 4$.
verified: the printed answer passed a computed check
How it was checked
- identity: passes a**2*b**2 - 2*a*b + 4