Special Products and Factoring

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A short course on special products and factoring usually given to high school or first year college students.

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Special Products and Factoring:

Special Products and Factoring

Special Products:

Special Products Product of a monomial and a binomial a(x + y) = ax + ay

Special Products:

Special Products Binomial Square (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2 The product of a binomial square is called PERFECT SQUARE TRINOMIAL

Special Products:

Special Products Binomial Square (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 – 2ab + b 2 Ex. (2x + 3) 2 = (2x) 2 + 2(2x)(3) + 3 2 = 4x 2 + 12x + 9

Special Products:

Special Products Product of the Sum and difference of two terms (a - b)(a + b) = a 2 - b 2 The product of the sum and difference of two terms is called: DIFFERENCE OF TWO SQUARES

Special Products:

Special Products Binomial Cube (x + y) 3 = x 3 + 3x 2 y + 3xy 2 + y 3 (x - y) 3 = x 3 - 3x 2 y + 3xy 2 - y 3

Factoring:

Factoring Common monomial factor ax + ay = a(x + y) Ex. 3x 2 + 6 - common factor is 3 3(x 2 + 2)

Factoring:

Factoring Difference of two squares x 2 - y 2 = (x) 2 – (y) 2 = (x + y)(x – y) Ex. 9x 2 – 4 = (3x) 2 – (2) 2 = [(3x) + 2][(3x) – 2] = (3x + 2)(3x – 2)

Factoring:

Factoring Perfect square trinomial x 2 + 2xy + y 2 = (x + y) 2 = (x + y)(x + y) x 2 - 2xy + y 2 = (x – y) 2 = (x – y)(x – y) Characteristics of a perfect square trinomial: The first term x 2 is a perfect square The last term is also a perfect square The middle term 2xy is twice the product of x and y The sign of the middle term determines the sign of the binomial

Factoring:

Factoring Perfect square trinomial x 2 + 2xy + y 2 = (x + y) 2 = (x + y)(x + y) x 2 - 2xy + y 2 = (x – y) 2 = (x – y)(x – y) Ex. 9x 2 – 6xy + y 2 9x 2 = (3x) 2 ; (y) 2 2(3x)(y) = 6xy 9x 2 – 6xy + y 2 = (3x – y) 2

Factoring:

Factoring Sum and difference of cubes x 3 + y 3 = (x) 3 + (y) 3 = (x + y)(x 2 – xy + y 2 ) x 3 - y 3 = (x) 3 - (y) 3 = (x - y)(x 2 + xy + y 2 )

Factoring:

Factoring Factoring by grouping ax + ay + bx + by = (ax + ay) + ( bx + by) = a(x + y) + b(x + y) = (a + b)(x +y)

Factoring:

Factoring Factoring a general trinomial ax 2 + bx + c How to test whether a given trinomial is factorable: Obtain the product ac. List down the factor pairs of the product ac. Choose among the factor pairs of ac that pair that will give a sum equal to the middle term b. * If no such pair gives a sum equal to the middle term b, then it is not factorable

Factoring:

Factoring Factoring a general trinomial ax 2 + bx + c Ex. 6x 2 + 5x – 21 5x

Factoring:

Factoring Factoring a general trinomial ax 2 + bx + c Try: