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8.5 – Factoring Differences of Squares
Recall:
Recall: Product of a Sum & a Difference
Recall: Product of a Sum & a Difference
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b)
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!*a and b MUST be perfect squares!
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!*a and b MUST be perfect squares!
Ex. 1 Factor each binomial below.a. n2 – 25
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!*a and b MUST be perfect squares!
Ex. 1 Factor each binomial below.a. n2 – 25 (n )(n )
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!*a and b MUST be perfect squares!
Ex. 1 Factor each binomial below.a. n2 – 25 (n + )(n – )
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!*a and b MUST be perfect squares!
Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!*a and b MUST be perfect squares!
Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)b. 36x2 – 49y2
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!*a and b MUST be perfect squares!
Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)b. 36x2 – 49y2
( x + )( x – )
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!*a and b MUST be perfect squares!
Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)b. 36x2 – 49y2
(6x + )(6x – )
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!*a and b MUST be perfect squares!
Ex. 1 Factor each binomial below.a. n2 – 25 (n + 5)(n – 5)b. 36x2 – 49y2
(6x + 7y)(6x – 7y)
Ex. 2 Factor each polynomial below.a. 48a3 – 12a
Ex. 2 Factor each polynomial below.a. 48a3 – 12a 12a( )
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a( a + )( a – )
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a( a + )( a – )
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + )(2a – )
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + )(2a – )
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
Ex. 2 Factor each polynomial below.a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1) b. 2x4 – 162
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2( )
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
2(x2 + )(x2 – )
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
2(x2 + 9)(x2 – 9)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
2(x2 + 9)(x2 – 9)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
2(x2 + 9)(x2 – 9)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
2(x2 + 9)(x2 – 9)
2(x2 + 9)(x + )(x – )
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
2(x2 + 9)(x2 – 9)
2(x2 + 9)(x + )(x – )
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
2(x2 + 9)(x2 – 9)
2(x2 + 9)(x + 3)(x – 3)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
2(x2 + 9)(x2 – 9)
2(x2 + 9)(x + 3)(x – 3)
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + )(p – ) = 0
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + 1)(p – 1) = 0
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + 1)(p – 1) = 0
p + 1 = 0 p – 1 = 0
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + 1)(p – 1) = 0
p + 1 = 0 p – 1 = 0
- 1 - 1 + 1 + 1
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + 1)(p – 1) = 0
p + 1 = 0 p – 1 = 0
- 1 - 1 + 1 + 1
p = -1 p = 1
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + 1)(p – 1) = 0
p + 1 = 0 p – 1 = 0
- 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x
Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0
(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x - 50x -50x
Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0
(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x - 50x -50x
18x3 – 50x = 0
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + 1)(p – 1) = 0
p + 1 = 0 p – 1 = 0
- 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x
- 50x -50x
18x3 – 50x = 0
2x(9x2 – 25) = 0
Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0
(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0
2x( x + )( x – ) = 0
Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0
(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0
2x(3x + )(3x – ) = 0
Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0
(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0
2x(3x + 5)(3x – 5) = 0
Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0
(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0
2x(3x + 5)(3x – 5) = 0 2x = 0
Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0
(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0
2x(3x + 5)(3x – 5) = 0 2x = 03x + 5 = 0
Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0
(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0
2x(3x + 5)(3x – 5) = 0 2x = 03x + 5 = 0 3x – 5 = 0
Ex. 3 Solve each equation by factoring.a. p2 – 1 = 0
(p + 1)(p – 1) = 0p + 1 = 0 p – 1 = 0 - 1 - 1 + 1 + 1
p = -1 p = 1
b. 18x3 = 50x - 50x -50x 18x3 – 50x = 02x(9x2 – 25) = 0
2x(3x + 5)(3x – 5) = 0 2x = 03x + 5 = 0 3x – 5 = 0 x = 0 x = -5/3 x = 5/3
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