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Factoring Differences of Squares

Factoring Differences of Squares. Multiplying Conjugates. The following pairs of binomials are called conjugates . Notice that they all have the same terms, only the sign between them is different. (3x + 6). and. (3x - 6). (r - 5). and. (r + 5). (2b - 1). and. (2b + 1). (x 2 + 5).

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Factoring Differences of Squares

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  1. Factoring Differences of Squares

  2. Multiplying Conjugates The following pairs of binomials are called conjugates. Notice that they all have the same terms, only the sign between them is different. (3x + 6) and (3x - 6) (r - 5) and (r + 5) (2b - 1) and (2b + 1) (x2 + 5) and (x2 - 5)

  3. x2 x x x x -x -x -x -x Multiplying Conjugates Multiply: (x + 4)(x – 4) using algebra tiles. x + 4 x - 4 = x • x + x • (-4) + 4 • x + 4 • (-4) FOIL: (x + 4)(x – 4) = x2 + (-4x) + 4x + (-16)

  4. x + 4 x2 x - 4 x x x x -x -x -x -x Multiplying Conjugates Multiply: (x + 4)(x – 4) using algebra tiles. Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs! = x • x + x • (-4) + 4 • x + 4 • (-4) FOIL: (x + 4)(x – 4) = x2 + (-4x) + 4x + (-16)

  5. x2 x x x -x -x -x Multiplying Conjugates Multiply: (x + 4)(x – 4) using algebra tiles. x + 4 x - 4 Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs! = x • x + x • (-4) + 4 • x + 4 • (-4) FOIL: (x + 4)(x – 4) = x2 + (-4x) + 4x + (-16)

  6. x2 x x -x -x Multiplying Conjugates Multiply: (x + 4)(x – 4) using algebra tiles. x + 4 x - 4 Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs! = x • x + x • (-4) + 4 • x + 4 • (-4) FOIL: (x + 4)(x – 4) = x2 + (-4x) + 4x + (-16)

  7. x2 x -x Multiplying Conjugates Multiply: (x + 4)(x – 4) using algebra tiles. x + 4 x - 4 Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs! = x • x + x • (-4) + 4 • x + 4 • (-4) FOIL: (x + 4)(x – 4) = x2 + (-4x) + 4x + (-16)

  8. x2 Multiplying Conjugates Multiply: (x + 4)(x – 4) using algebra tiles. x + 4 x - 4 Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs! = x • x + x • (-4) + 4 • x + 4 • (-4) FOIL: (x + 4)(x – 4) = x2 + (-4x) + 4x + (-16)

  9. x2 Multiplying Conjugates Multiply: (x + 4)(x – 4) using algebra tiles. x + 4 x - 4 = x2 + (-16) = x • x + x • (-4) + 4 • x + 4 • (-4) FOIL: (x + 4)(x – 4) = x2 + (-4x) + 4x + (-16)

  10. x2 Multiplying Conjugates Multiply: (x + 4)(x – 4) using algebra tiles. x + 4 x - 4 Opposite terms also add up to zero (or cancel). Cancel out any opposite pairs! = x • x + x • (-4) + 4 • x + 4 • (-4) FOIL: (x + 4)(x – 4) = x2 + (-4x) + 4x + (-16)

  11. x2 Multiplying Conjugates Multiply: (x + 4)(x – 4) using algebra tiles. x + 4 x - 4 Opposite terms also add up to zero (or cancel). Cancel out any opposite pairs! = x • x + x • (-4) + 4 • x + 4 • (-4) FOIL: (x + 4)(x – 4) = x2 + (-4x) + 4x + (-16) = x2 + 0 + (-16) = x2 + (-16)

  12. Multiplying Conjugates When we multiply any conjugate pairs, the middle terms always cancel and we end up with a binomial. (3x + 6)(3x - 6) = 9x2 - 36 (r - 5)(r + 5) = r2 - 25 = 4b2 - 1 (2b - 1)(2b + 1)

  13. A MINUS between! Difference of Squares Binomials that look like this are called a Difference of Squares: Only TWO terms (a binomial) 9x2 - 36 The first term is a Perfect Square! The second term is a Perfect Square!

  14. A Difference of Squares! A Conjugate Pair! Factor a Difference of Squares:

  15. Factor a Difference of Squares: = (x + 8)(x - 8) Example: Factor x2 - 64 x2 = x • x 64 = 8 • 8 = (3t + 5)(3t - 5) Example: Factor 9t2 - 25 9t2 = 3t • 3t 25 = 5 • 5

  16. A Sum of Squares? A Sum of Squares, like x2 + 64, can NOT be factored! It is a PRIME polynomial.

  17. Practice Factor each polynomial. 1) x2 - 81 2) r2 - 100 3) 16 - a2 4) 9a2 - 16 5) 16x2 - 1

  18. Practice - Answers Factor each polynomial. 1) x2 - 81 = (x + 9)(x - 9) 2) r2 - 100 = (r + 10)(r - 10) 3) 16 - a2 = (4 + a)(4 - a) 4) 9a2 - 16 = (3a + 4)(3a - 4) 5) 16x2 - 1 = (4x + 1)(4x - 1)

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