![Page 1: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/1.jpg)
8.5 – Factoring Differences of Squares
![Page 2: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/2.jpg)
Recall:
![Page 3: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/3.jpg)
Recall: Product of a Sum & a Difference
![Page 4: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/4.jpg)
Recall: Product of a Sum & a Difference
![Page 5: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/5.jpg)
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b)
![Page 6: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/6.jpg)
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
![Page 7: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/7.jpg)
Recall: Product of a Sum & a Difference (8.6)(a + b)(a – b) = a2 – b2
*Use in reverse to factor!
![Page 8: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/8.jpg)
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!
![Page 9: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/9.jpg)
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
![Page 10: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/10.jpg)
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 )
![Page 11: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/11.jpg)
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 – )
![Page 12: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/12.jpg)
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)
![Page 13: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/13.jpg)
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
![Page 14: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/14.jpg)
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 – )
![Page 15: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/15.jpg)
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 – )
![Page 16: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/16.jpg)
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)
![Page 17: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/17.jpg)
Ex. 2 Factor each polynomial below.a. 48a3 – 12a
![Page 18: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/18.jpg)
Ex. 2 Factor each polynomial below.a. 48a3 – 12a 12a( )
![Page 19: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/19.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
![Page 20: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/20.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
![Page 21: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/21.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
![Page 22: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/22.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a( a + )( a – )
![Page 23: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/23.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a( a + )( a – )
![Page 24: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/24.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + )(2a – )
![Page 25: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/25.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + )(2a – )
![Page 26: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/26.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
![Page 27: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/27.jpg)
Ex. 2 Factor each polynomial below.a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1) b. 2x4 – 162
![Page 28: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/28.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2( )
![Page 29: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/29.jpg)
Ex. 2 Factor each polynomial below.
a. 48a3 – 12a
12a(4a2 – 1)
12a(2a + 1)(2a – 1)
b. 2x4 – 162
2(x4 – 81)
![Page 30: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/30.jpg)
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 – )
![Page 31: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/31.jpg)
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)
![Page 32: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/32.jpg)
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)
![Page 33: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/33.jpg)
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)
![Page 34: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/34.jpg)
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 – )
![Page 35: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/35.jpg)
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 – )
![Page 36: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/36.jpg)
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)
![Page 37: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/37.jpg)
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)
![Page 38: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/38.jpg)
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
![Page 39: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/39.jpg)
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + )(p – ) = 0
![Page 40: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/40.jpg)
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + 1)(p – 1) = 0
![Page 41: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/41.jpg)
Ex. 3 Solve each equation by factoring.
a. p2 – 1 = 0
(p + 1)(p – 1) = 0
p + 1 = 0 p – 1 = 0
![Page 42: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/42.jpg)
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
![Page 43: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/43.jpg)
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
![Page 44: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/44.jpg)
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
![Page 45: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/45.jpg)
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
![Page 46: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/46.jpg)
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
![Page 47: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/47.jpg)
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
![Page 48: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/48.jpg)
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
![Page 49: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/49.jpg)
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
![Page 50: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/50.jpg)
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
![Page 51: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/51.jpg)
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
![Page 52: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/52.jpg)
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
![Page 53: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/53.jpg)
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
![Page 54: 8.5 – Factoring Differences of Squares. Recall: Recall: Product of a Sum & a Difference](https://reader036.vdocuments.net/reader036/viewer/2022062421/56649e4c5503460f94b423e5/html5/thumbnails/54.jpg)
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