1. © William James Calhoun, 2001 9-6: Multiplying a Polynomial by a Monomial OBJECTIVES: You will multiply a polynomial by a monomial and simplify expressions involving polynomials. This section is essentially how to use the distributive property. The distributive property is a tool you already have in your mathematics toolbox. The difference is that you will now be faced with using the tools from the first half of this chapter while going through the distribution process.

2. © William James Calhoun, 2001 EXAMPLE 1: Find each product. A. 7b(4b 2 - 18)B. -3y 2 (6y 2 - 8y + 12) Use the distributive property first. 7b(4b 2 ) + 7b(-18) 7(4)(b)(b 2 ) + 7(-18)(b) Use the properties from 9-1 to 9-4. Re-arrange the terms. Multiply constants and use exponents for variables. 28b 3 + -126b Combine like terms and simplify. 28b 3 - 126b -3y 2 (6y 2 ) + (-3y 2 )(-8y) + (-3y 2 )(12) (-3)(6)(y 2 )(y 2 ) + (-3)(-8)(y 2 )(y) + (-3)(12)(y 2 ) Use the properties from 9-1 to 9-4. Re-arrange the terms. Multiply constants and use exponents for variables. -18y 4 + 24y 3 + -36y 2 Combine like terms and simplify. -18y 4 + 24y 3 - 36y 2 9-6: Multiplying a Polynomial by a Monomial

3. © William James Calhoun, 2001 EXAMPLE 2: Simplify -3pq(p 2 q + 2p - 3p 2 q). This problem is a bit unusual. There are like terms in the parenthesis which have not been combined yet. It will be easier to combine them first, then do the distribution. -3pq(1p 2 q - 3p 2 q + 2p) -3pq(-2p 2 q + 2p) -3pq(-2p 2 q) + (-3pq)(2p) -3(-2)(p)(p 2 )(q)(q) + (-3)(2)(p)(p)(q) Use the properties from 9-1 to 9-4. Re-arrange the terms. Multiply constants and use exponents for variables. 6p 3 q 2 + -6p 2 q Combine like terms and simplify. 6p 3 q 2 - 6p 2 q 9-6: Multiplying a Polynomial by a Monomial

4. © William James Calhoun, 2001 EXAMPLE 3: Solve the equation: x(x + 3) + 7x - 5 = x(8 + x) - 9x + 14. Step back to the rules for solving equations - the full set of rules. (1) Distribute: x(x) + x(3) + 7x - 5 = x(8) + x(x) - 9x + 14 x 2 + 3x + 7x - 5 = 8x + x 2 - 9x + 14 (2) Combine like terms on each side. x 2 + 10x - 5 = x 2 - 1x + 14 (3) Move all variables to same side. x 2 + 10x - 5 = x 2 - 1x + 14 -x 2 10x - 5 = - 1x + 14 +1x + 1x 11x - 5 = 14 (4) Add/subtract from both sides. 11x - 5 = 14 +5 11x = 19 (5) Multiply/divide on both sides. 11x = 19 11 9-6: Multiplying a Polynomial by a Monomial

5. © William James Calhoun, 2001 9-6: Multiplying a Polynomial by a Monomial HOMEWORK Page 532 #17 - 35 odd