9-8 notes
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Factoring Sums and Differences of PowersTRANSCRIPT
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