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) (x 2 + 5) and (x 2 - 5)

3. Multiply: (x + 4)(x – 4) using algebra tiles. Multiplying Conjugates x + 4 x-4x-4 x2x2 x -x xxx FOIL: (x + 4)(x – 4) = x x + x (-4) + 4 x + 4 (-4) = x 2 + (-4x) + 4x + (-16)

4. Multiply: (x + 4)(x – 4) using algebra tiles. Multiplying Conjugates x + 4 x-4x-4 x2x2 x -x xxx FOIL: (x + 4)(x – 4) = x x + x (-4) + 4 x + 4 (-4) = x 2 + (-4x) + 4x + (-16) Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

5. Multiply: (x + 4)(x – 4) using algebra tiles. Multiplying Conjugates x + 4 x-4x-4 x2x2 x -x xx FOIL: (x + 4)(x – 4) = x x + x (-4) + 4 x + 4 (-4) = x 2 + (-4x) + 4x + (-16) Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

6. Multiply: (x + 4)(x – 4) using algebra tiles. Multiplying Conjugates x + 4 x-4x-4 x2x2 x -x x FOIL: (x + 4)(x – 4) = x x + x (-4) + 4 x + 4 (-4) = x 2 + (-4x) + 4x + (-16) Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

7. Multiply: (x + 4)(x – 4) using algebra tiles. Multiplying Conjugates x + 4 x-4x-4 x2x2 x -x FOIL: (x + 4)(x – 4) = x x + x (-4) + 4 x + 4 (-4) = x 2 + (-4x) + 4x + (-16) Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

8. Multiply: (x + 4)(x – 4) using algebra tiles. Multiplying Conjugates x + 4 x-4x-4 x2x2 FOIL: (x + 4)(x – 4) = x x + x (-4) + 4 x + 4 (-4) = x 2 + (-4x) + 4x + (-16) Opposite tiles add up to zero (or cancel). Cancel out any opposite pairs!

9. Multiply: (x + 4)(x – 4) using algebra tiles. Multiplying Conjugates x + 4 x-4x-4 x2x2 FOIL: (x + 4)(x – 4) = x x + x (-4) + 4 x + 4 (-4) = x 2 + (-4x) + 4x + (-16) = x 2 + (-16)

10. Multiply: (x + 4)(x – 4) using algebra tiles. Multiplying Conjugates x + 4 x-4x-4 x2x2 FOIL: (x + 4)(x – 4) = x x + x (-4) + 4 x + 4 (-4) = x 2 + (-4x) + 4x + (-16) Opposite terms also add up to zero (or cancel). Cancel out any opposite pairs!

11. Multiply: (x + 4)(x – 4) using algebra tiles. Multiplying Conjugates x + 4 x-4x-4 x2x2 FOIL: (x + 4)(x – 4) = x x + x (-4) + 4 x + 4 (-4) = x 2 + (-4x) + 4x + (-16) Opposite terms also add up to zero (or cancel). Cancel out any opposite pairs! = x 2 + 0 + (-16) = x 2 + (-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) (r - 5)(r + 5) (2b - 1)(2b + 1) = 9x 2 - 36 = r 2 - 25 = 4b 2 - 1

13. Difference of Squares Binomials that look like this are called a Difference of Squares: 9x 2 - 36 The first term is a Perfect Square! The second term is a Perfect Square! Only TWO terms (a binomial) A MINUS between!

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

15. Factor a Difference of Squares: Example: Factor x 2 - 64 x 2 = x x 64 = 8 8 = (x + 8)(x - 8) Example: Factor 9 t 2 - 25 9t 2 = 3t 3t 25 = 5 5 = (3t + 5)(3t - 5)

16. A Sum of Squares? A Sum of Squares, like x 2 + 64, can NOT be factored! It is a PRIME polynomial.

17. Factor each polynomial. Practice 1) x 2 - 81 2) r 2 - 100 3) 16 - a 2 4) 9a 2 - 16 5) 16x 2 - 1

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