Section 9-8Factoring Sums and Differences of Powers

Warm-upFactor over the set of polynomials with the indicated

coefficients:

x4 −1

1. Real coefficients 2. Complex coefficients

Warm-upFactor over the set of polynomials with the indicated

coefficients:

x4 −1

1. Real coefficients 2. Complex coefficients

x4 −1

Warm-upFactor over the set of polynomials with the indicated

coefficients:

x4 −1

1. Real coefficients 2. Complex coefficients

x4 −1

(x2 −1)(x 2 +1)

Warm-upFactor over the set of polynomials with the indicated

coefficients:

x4 −1

1. Real coefficients 2. Complex coefficients

x4 −1

(x2 −1)(x 2 +1)

(x −1)(x +1)(x 2 +1)

Warm-upFactor over the set of polynomials with the indicated

coefficients:

x4 −1

1. Real coefficients 2. Complex coefficients

x4 −1

(x2 −1)(x 2 +1)

(x −1)(x +1)(x 2 +1)

x4 −1

Warm-upFactor over the set of polynomials with the indicated

coefficients:

x4 −1

1. Real coefficients 2. Complex coefficients

x4 −1

(x2 −1)(x 2 +1)

(x −1)(x +1)(x 2 +1)

x4 −1

(x2 −1)(x 2 +1)

Warm-upFactor over the set of polynomials with the indicated

coefficients:

x4 −1

1. Real coefficients 2. Complex coefficients

x4 −1

(x2 −1)(x 2 +1)

(x −1)(x +1)(x 2 +1)

x4 −1

(x2 −1)(x 2 +1)

(x2 −1)(x 2 − (−1))

Warm-upFactor over the set of polynomials with the indicated

coefficients:

x4 −1

1. Real coefficients 2. Complex coefficients

x4 −1

(x2 −1)(x 2 +1)

(x −1)(x +1)(x 2 +1)

x4 −1

(x2 −1)(x 2 +1)

(x −1)(x +1)(x − i)(x + i) (x

2 −1)(x 2 − (−1))

Example 1Find the four fourth roots of 16.

Example 1Find the four fourth roots of 16.

x4 =16

Example 1Find the four fourth roots of 16.

x4 =16

x4 −16 = 0

Example 1Find the four fourth roots of 16.

x4 =16

x4 −16 = 0

(x2 − 4)(x 2 + 4) = 0

Example 1Find the four fourth roots of 16.

x4 =16

x4 −16 = 0

(x2 − 4)(x 2 + 4) = 0

(x − 2)(x + 2)(x − 2i)(x + 2i) = 0

Example 1Find the four fourth roots of 16.

x4 =16

x4 −16 = 0

(x2 − 4)(x 2 + 4) = 0

(x − 2)(x + 2)(x − 2i)(x + 2i) = 0

x = ±2,±2i

Example 1Find the four fourth roots of 16.

x4 =16

x4 −16 = 0

(x2 − 4)(x 2 + 4) = 0

(x − 2)(x + 2)(x − 2i)(x + 2i) = 0

x = ±2,±2i

The roots are ±2,±2i

Sums and Differences of Cubes Theorem

Sums and Differences of Cubes Theorem

x3 + y3 = (x + y )(x 2 − xy + y 2 )

Sums and Differences of Cubes Theorem

x3 + y3 = (x + y )(x 2 − xy + y 2 )

x3 − y3 = (x − y )(x 2 + xy + y 2 )

Example 2Factor over the set of polynomials with rational

coefficients.

x3 − 64y12

Example 2Factor over the set of polynomials with rational

coefficients.

x3 − 64y12

(x − 4y 4 )(x 2 + 4xy 4 +16y 8 )

Sums and Differences of Odd Powers Theorem

Sums and Differences of Odd Powers TheoremFor all x and y and for all positive integers n:

Sums and Differences of Odd Powers TheoremFor all x and y and for all positive integers n:

xn + y n = (x + y )(xn−1 − xn−2y + xn−3y 2 − ... − xy n−2 + y n−1)

Sums and Differences of Odd Powers TheoremFor all x and y and for all positive integers n:

xn + y n = (x + y )(xn−1 − xn−2y + xn−3y 2 − ... − xy n−2 + y n−1)

xn − y n = (x − y )(xn−1 + xn−2y + xn−3y 2 + ... + xy n−2 + y n−1)

Example 3Factor over the set of polynomials with rational

coefficients.

t7 − w7

Example 3Factor over the set of polynomials with rational

coefficients.

t7 − w7

(t − w )(t6 + t5w + t4w 2 + t3w3 + t2w 4 + tw5 + w6 )

Example 4Factor over the set of polynomials with rational

coefficients.

m6 + n6

Example 4Factor over the set of polynomials with rational

coefficients.

m6 + n6

= (m2 )3 + (n2 )3

Example 4Factor over the set of polynomials with rational

coefficients.

m6 + n6

= (m2 )3 + (n2 )3

= (m2 + n2 )(m4 − m2n2 + y 4 )

Example 5Factor completely over the set of polynomials with rational

coefficients.

x10 − y10

Example 5Factor completely over the set of polynomials with rational

coefficients.

x10 − y10

= (x5 )2 − (y 5 )2

Example 5Factor completely over the set of polynomials with rational

coefficients.

x10 − y10

= (x5 )2 − (y 5 )2

= (x5 − y 5 )(x5 + y 5 )

Example 5Factor completely over the set of polynomials with rational

coefficients.

x10 − y10

= (x5 )2 − (y 5 )2

= (x5 − y 5 )(x5 + y 5 )

= (x − y )(x 4 + x3y + x 2y 2 + xy3 + y 4 )(x + y )(x 4 − x3y + x 2y 2 − xy3 + y 4 )

Homework

Homework

p. 605 #1-19