Multiplying Conjugates The following pairs of binomials are called conjugates. Notice that they all have the same terms, only the sign between them is.

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Presentation transcript:

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)

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)

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!

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!

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!

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!

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!

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)

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!

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 (-16) = x 2 + (-16)

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 = r = 4b 2 - 1

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

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

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

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

Factor each polynomial. Practice 1) x ) r ) 16 - a 2 4) 9a ) 16x 2 - 1

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