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Section 8.2

Section 8.2. Factoring Using the Distributive Property. Factor polynomials by using the Distributive Property. Solve quadratic equations of the form ax 2 + bx = 0. factoring. factoring by grouping Zero Products Property roots. Factor by Using the Distributive Property.

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Section 8.2

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  1. Section 8.2 Factoring Using the Distributive Property

  2. Factor polynomials by using the Distributive Property. • Solve quadratic equations of the form ax2 + bx = 0. • factoring • factoring by grouping • Zero Products Property • roots

  3. Factor by Using the Distributive Property In Ch.7, you used the distributive property to multiply a polynomial by a monomial. 2a(6a + 8) = 2a(6a) + 2a(8) = 12a² + 16a You can reverse this process to express a polynomial as the product of a monomial factor and a polynomial factor. 12a² + 16a = 2a(6a) + 2a(8) = 2a(6a + 8) Factoring a polynomial means to find its completely factored form.

  4. Use the Distributive Property A. Use the Distributive Property to factor 15x + 25x2. First, find the GCF of 15x + 25x2. 15x = 3 ● 5 ● x Factor each monomial. 25x2 = 5 ● 5 ● x ● x Circle the common prime factors. GCF: 5 ● x or 5x Write each term as the product of the GCF and its remaining factors. Then use the Distributive Property to factor out the GCF. 15x + 25x2 = 5x(3) + 5x(5 ● x) Rewrite each term using the GCF.

  5. Use the Distributive Property = 5x(3) + 5x(5x) Simplify remaining factors. = 5x(3 + 5x) Distributive Property Answer: 5x(3 + 5x)

  6. 12xy = 2 ● 2 ● 3 ● x ● y Factor each monomial. 24xy2 = 2 ● 2 ● 2 ● 3 ● x ● y ● y 30x2y4 = 2 ● 3 ● 5 ● x ● x● y ● y ● y ● y Use the Distributive Property B. Use the Distributive Property to factor 12xy + 24xy2 – 30x2y4. Circle the common prime factors. GCF: 2 ● 3 ● x ● y or 6xy 12xy + 24xy2 – 30x2y4 = 6xy(2) + 6xy(4y) + 6xy(–5xy3) Rewrite each term using the GCF.

  7. Use the Distributive Property = 6xy(2 + 4y – 5xy3) Distributive Property Answer: The factored form of 12xy + 24xy2 – 30x2y4 is 6xy(2 + 4y – 5xy3).

  8. Using the Distributive Property to factor polynomials having four or more terms is called factoring by groupingbecause pairs of terms are grouped together and factored.

  9. Use Grouping Factor 2xy + 7x – 2y – 7. 2xy + 7x – 2y – 7 = (2xy – 2y) + (7x – 7) Group terms with common factors. = 2y(x – 1) + 7(x – 1) Factor the GCF from each grouping. = (x – 1)(2y + 7) Distributive Property Answer: (x – 1)(2y + 7)

  10. Recognizing binomials that are additive inverses is often helpful when • factoring by grouping. • For example, 7 - y and y – 7 are additive inverses. • By rewriting 7 - yas -1(y – 7), factoring by grouping is possible

  11. Use the Additive Inverse Property Factor 15a – 3ab + 4b – 20. 15a – 3ab + 4b – 20 = (15a – 3ab) + (4b – 20)Group terms with common factors. = 3a(5 – b) + 4(b – 5)Factor GCF from each grouping. = 3a(–1)(b – 5) + 4(b – 5)5 – b = –1(b – 5) = –3a(b – 5) + 4(b – 5)3a(–1) = –3a Answer: = (b – 5)(–3a + 4) Distributive Property

  12. Some equations can be solved by factoring. Consider the following: 6(0) = 0 0(-3) = 0 (5 – 5)(0) = 0 -2(-3 + 3) = 0 Notice that in each case, at least one of the factors is zero. The solutions of an equation are called the roots of the equation.

  13. Solve an Equation A. Solve (x – 2)(4x – 1) = 0. Check the solution. If (x – 2)(4x – 1) = 0, then according to the Zero Product Property, either x – 2 = 0 or 4x – 1 = 0. (x – 2)(4x – 1) = 0 Original equation x – 2 = 0 or 4x – 1 = 0 Set each factor equal to zero. x = 2 4x = 1 Solve each equation.

  14. Homework Assignment #43 8.2 Skills Practice Sheet

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