Chapters 8 and 9 Greatest Common Factors & Factoring by Grouping

Chapters 8 and 9 Greatest Common Factors & Factoring by Grouping • Definitions • Factor, Factoring, Prime Polynomial • Common Factor of 2 or more terms • Factoring a Monomial into two factors • Identifying Common Monomial Factors • Factoring Out Common Factors • Arranging a 4 Term Polynomial into Groups • Factoring Out Common Binomials

What’s a Polynomial Factor? product = (factor)(factor)(factor) … (factor) Factoring is the reverse of multiplication. 84 is a product that can be expressed by many different factorizations: 84 = 2(42) or 84 = 7(12) or 84 = 4(7)(3) or 84 = 2(2)(3)(7) Only one example, 84 = 2(2)(3)(7), shows 84 as the product of prime integers. Always try to factor a polynomial into prime polynomials

Factoring Monomials • 12x3 also can be expressed in many ways:12x3 = 12(x3) 12x3 = 4x2(3x) 12x3 = 2x(6x2) • Usually, we only look for two factors – You try: • 4a = • 2(2a) or 4(a) • x3 = • x(x2)orx2(x) • 14y2 = • 14(y2)or14y(y)or7(2y2)or7y(2y)ory(14y) • 43x5 = • 43(x5)or43x(x4)orx3(43x2)or43x2(x3)or…

Common Factors of Polynomials • When a polynomial has 2 or more terms, it may have common factors • By definition, a common factor must divide evenly into every term • For x2 + 3x the only common factor is x , so • x2 + 3x = x·x + x·3 = x(? + ?) = x(x + 3) • For 8y2 + 12y – 20 a common factor is 2, so • 8y2 + 12y – 20 = 2(? + ? – ?) =2(4y2 + 6y – 10) • Check factoring by multiplying: • 2(4y2 + 6y – 10) = 8y2 + 12y – 20

The Greatest Common Factor of Polynomials • The greatestcommon factor (or GCF) is the largest monomial that can divide evenly into every term • Looking for common factors in 2 or more terms … is always the first step in factoring polynomials • Remember a(b + c) = ab + ac (distributive law) • Consider that a is a common factor of ab + ac • If we find a polynomial has form ab + ac we can factor it into a(b + c) • For 3x2 + 3x the greatest common factor is 3x , so • 3x2 + 3x = 3x·x + 3x·1 = 3x(? + ?) = 3x(x + 1) • Another example: 8y2 + 12y – 20 • The GCF is 4 – Divide each term by 4 • 8y2 + 12y – 20 = 4(? + ? – ?) = 4(2y2 + 3y – 5) • Check by multiplying: 4(2y2) + 4(3y) – 4(5) = 8y2 + 12y – 20

Practice: Find the Greatest Common Monomial Factor • 7a – 21 = • 7(? – ?) = • 7(a – 3) • 19y3 + 3y = • y(? + ?) = • y(19y2 + 3) • 8x2 + 14x – 4 = • 2(? + ? – ?) = • 2(4x2 + 7x – 2) • 4y2 + 6y = • 2y(? + ?) = • 2y(2y+ 3)

Find the Greatest Common Factor • 18y5 – 12y4 + 6y3 = • 6y3(? – ? + ?) = • 6y3(3y2 – 2y + 1) • 21x2 – 42xy + 28y2 = • 7(? – ? + ?) = • 7(3x2 – 6xy + 4y2) • 22x3 – 110xy2 = • 22x(? – ?) = • 22x(x2 – 5y2) • 7x2 – 11xy + 13y2 = • No common factor exists

Introduction to Factoring by Grouping:Factoring Out Binomials • x2(x + 7) + 3(x + 7) = • (x + 7)(? + ?) = • (x + 7)(x2 + 3) • y3(a + b) – 2(a + b) = • (a + b)(? – ?) = • (a + b)(y3 – 2)

Practice:Factoring Out Binomials • You try: 2x2(x – 1) + 6x(x – 1) – 17(x – 1) = • (x – 1)(? + ? – ?) • (x – 1)(2x2 + 6x – 17) • y2(2y – 5) + x2(2y – 5) = • (2y – 5)(? + ?) • (2y – 5)(y2 + x2) • 5x2(xy + 1) + 6y(xy – 1) = • No common factors

Factoring by Grouping • For polynomials with 4 terms: • Arrange the terms in the polynomial into 2 groups • Factor out the common monomials from each group • If the binomial factors produced are either identical or opposite, complete the factorization • Example: 2c – 2d + cd – d2 • 2(c – d) + d(c – d) • (c – d)(2 + d)

Factor by Grouping • 8t3 + 2t2 – 12t – 3 • 2t2(4t + 1)– 3(4t + 1) • (4t + 1)(2t2 – 3)

Factor by Grouping • 4x3 – 6x2 – 6x + 9 • 2x2(2x – 3) – 3(2x – 3) • (2x – 3)(2x2 – 3)

Factor by Grouping • y4 – 2y3 – 12y – 3 • y3(y – 2) – 3(4y – 1) • Oops –not factorablevia grouping

Grouping Unusual Polynomials • x3 – 7x2 + 6x + x2y – 7xy + 6y • x(x2 – 7x + 6) + y(x2 – 7x + 6) • (x2 – 7x + 6)(x + y) • (x –1)(x – 6)(x + y)

What Next? • Section 5.6 – Factoring Trinomials

Chapters 8 and 9 Greatest Common Factors & Factoring by Grouping