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Sect. 5.3 Common Factors & Factoring by Grouping

Sect. 5.3 Common Factors & Factoring by Grouping. Definitions Factor Common Factor of 2 or more terms Factoring a Monomial into two factors Identifying Common Monomial Factors Factoring Out Common Factors Arranging 4 Term Polynomials into 2 Groups Factoring Out Common Binomials.

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Sect. 5.3 Common Factors & Factoring by Grouping

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  1. Sect. 5.3 Common Factors & Factoring by Grouping • Definitions • Factor • Common Factor of 2 or more terms • Factoring a Monomial into two factors • Identifying Common Monomial Factors • Factoring Out Common Factors • Arranging 4 Term Polynomials into 2 Groups • Factoring Out Common Binomials 5.3

  2. What’s a Factor? product = (factor)(factor)(factor) … (factor) 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. Factoringis the reverse of multiplication. 5.3

  3. 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 • Your turn – factor these monomials into two factors: • 4a = • 2(2a)or4(a) • x3 = • x(x2) orx2(x) • 14y2 = • 14(y2) or14y(y)or7(2y2)or7y(2y)ory(14y)or … • 43x5 = • 43(x5)or43x(x4)orx3(43x2) or43x2(x3) or … 5.3

  4. Common Factors • Sometimes multi-termed polynomials can be factored • Looking for common factors in 2 or more terms … is the first step in factoring polynomials • Remember a(b + c) = ab + ac (distributive law) • Consider that a is a common factor of ab + acso we can factor ab + ac into a(b + c) • For x2 + 3x the only common factor is x , so • x2 + 3x = x(? + ?) = x(x + 3) • Another example: 4y2 + 6y – 10 • The common factor is 2 • 4y2 + 6y – 10 = 2(? + ? – ?) = 2(2y2 + 3y – 5) • Check by multiplying: 2(2y2) + 2(3y) – 2(5) = 4y2 + 6y – 10 5.3

  5. Find the Greatest Common Factor • 7a – 21 = • 7(? – ?) = • 7(a – 3) • 19x3 + 3x = • x(? + ?) = • x(19x2 + 3) • 18y3 – 12y2 + 6y = • 6y(? – ? + ?) = • 6y(3y2 – 2y + 1) 5.3

  6. 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) • You try: 2x2(x – 1) + 6x(x – 1) + 17(x – 1) = • (x – 1)(? + ? – ?) • (x – 1)(2x2 + 6x + 17) 5.3

  7. Factoring by Grouping • For polynomials with 4 terms: • Arrange the terms in the polynomial into 2 groupssuch thateach group has a common monomial factor • Factor out the common monomials from each group(the binomial factors produced will be either identical or opposites) • Factor out the common binomial factor • Example: 2c – 2d + cd – d2 • 2(c – d) + d(c – d) • (c – d)(2 + d) 5.3

  8. Factor by Grouping • 8t3 + 2t2 – 12t – 3 • 2t2(4t + 1)– 3(4t + 1) • (4t + 1)(2t2 – 3) • 4x3 – 6x2 – 6x + 9 • 2x2(2x – 3) – 3(2x – 3) • (2x – 3)(2x2 – 3) • y4 – 2y3 – 12y – 3 • y3(y – 2) – 3(4y + 1) • Oops –not factorable via grouping 5.3

  9. What Next? • Next time: Section 5.4 –Factoring Trinomials 5.3

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