Section 9-2 Multiply and Factor Polynomials SPI 12D: multiply two polynomials with each factor having no more than two terms Objectives: Multiply a polynomial by a monomial Factor a monomial from a polynomial Apply polynomials to a real-world situation Distributive Property: 5(a + 3) = 5a + 15 (5 ∙ a) + (5 ∙ 3) Multiply terms with exponents: Must have like bases Add the exponents 5x ∙ 2x = 10 x 1+1 = 10x 2 Combine like terms by adding and subtracting coefficients: Must have the same variable, raised to the same power (5x 2 and 32x 2 are like terms) (6x 7 and 7x 4 are NOT like terms)

Use the Definition of Area to Model Multiplying Polynomials What is the formula for Area? A = l x w x 1 Legend Find the area of the rectangle. x x x 1 3x + 1 xxxx 2x x2x2 x2x2 x2x2 x2x2 x2x2 x2x2 x x How many x 2 ‘s do you have? How many x ‘s do you have? The total area is: 6x 2 2x What is total area? 6x 2 + 2x Algebraic expression to model the problem. 2x(3x + 1) =2x(3x) + 2x(1) = 6x 2 + 2x

Simplify –2g 2 (3g 3 + 6g – 5). –2g 2 (3g 3 + 6g – 5) = –2g 2 (3g 3 ) –2g 2 (6g) –2g 2 (–5)Distributive Property = –6g – 12g g 2 Simplify = –6g 5 – 12g g 2 Simplify and write in standard form Use the Distributive Property to simplify an expression. sign (+ or - ) in front of monomial is part of the monomial outside term is distributed to each inside term remember to combine like terms remember when multiplying by like bases, add exponents Simplify Polynomials using the Distributive Property

Simplify –7h(3h 2 – 8h – 1). –7h(3h 2 – 8h – 1) =–7h(3h 2 ) + (-7h)(– 8h) + (-7h)( – 1) = –21h h 2 + 7h

Find the GCF of 2x x 2 – 6x. List the prime factors of each term. Identify the factors common to all terms. 2x 4 = 2 x x x x 10x 2 = 2 5 x x -6x = 2 3 x (-) The GCF is 2 x, or 2x, so………………….. Factoring a polynomial reverses the multiplication process. It undoes the Distributive Property To factor a monomial from a polynomial: 1. Find the GCF (Greatest common factor) prime numbers: only factors are the number itself and 1 2. Use GCF to factor each polynomial 2x(x 3 + 5x – 3) 2x ∙ x 3 2x ∙ 5x 2x ∙ - 3 Factor a Polynomial

Factor 4x x 2 – 16x. Step 1: Find the GCF. 4x 3 = 2 2 x x x 12x 2 = x x 16x = x Step 2: Factor out the GCF. 4x x 2 – 16x = 4x(x 2 ) + 4x(3x) + 4x(–4) = 4x(x 2 + 3x – 4) The GCF is 2 2 x, or 4x. Factor 6m 3 – 12m 2 – 24m. Step 1: Find the GCF. 6m 3 = 2 3 m m m 12m 2 = m m 24m = m Step 2: Factor out the GCF. 6m m 2 – 24m = 6m(m 2 ) - 6m(2m) – 6m(4) = 6m(m 2 – 2m – 4) The GCF is 2 3 m or 6m. Practice Factoring Polynomials

Building Models: Suppose you are an architect and you are building a model of a square castle shown as the white square. The moat of the castle is made of blue paper as shown in the diagram. 1. Find the area of the moat using the dimensions given in the diagram. 2. Write your answer in factored form. A of circle = π r 2 A of moat = π (4x) 2 – (2x) 2 A of moat = 16πx 2 – 4x 2 A of moat = 4x 2 (4π – 1)