2. © William James Calhoun, 2001 9-7: Multiplying Polynomials OBJECTIVES: The student will (1) use the FOIL method to multiply two binomials, and (2) multiply any two polynomials by using the distributive property. This section is again dealing with the distributive property. The difference is that you will be distributing more than one term in the same problem. The easiest of these types of problems is a binomial times a binomial where you must distribute both terms of the first binomial across both terms of the second binomial. There is a new tool for this easiest of cases.

3. © William James Calhoun, 2001 9-7: Multiplying Polynomials 9.7.1: FOIL Method for Multiplying Two Binomials To multiply two binomials, find the sum of the products of: F the First terms, O the Outer terms, I the Inner terms, L the Last terms. In the case of (7x + 2)(5x + 1) (7x)(1)(7x)(5x)(2)(5x)(2)(1)+++ First Outer Inner Last Simplify using the tools from 9-1 to 9-4. 35x 2 + 7x + 10x + 2 Combine like terms. 35x 2 + 17x + 2

4. © William James Calhoun, 2001 9-7: Multiplying Polynomials EXAMPLE 1: Find each product. A. (x - 4)(x + 9)B. (4x + 7)(3x - 8) Use 9.7.1: FOIL. First Outside Inside Last (x)(x) + (x)(9) + (-4)(x) + (-4)(9) Multiply each term out. x 2 + 9x + -4x + -36 Combine like terms and clean up the answer. x 2 + 5x - 36 Use 9.7.1: FOIL. First Outside Inside Last (4x)(3x) + (4x)(-8) + (7)(3x) + (7)(-8) Group each term. (4)(3)(x)(x) + (4)(-8)(x) + (7)(3)(x) + (7)(-8) Multiply each term out. 12x 2 + -32x + 21x + -56 Combine like terms and clean up the answer. 12x 2 - 11x - 56

5. © William James Calhoun, 2001 9-7: Multiplying Polynomials EXAMPLE 2: Find the product (2y + 5)(3y 2 - 8y + 7). Distribute the 2y and the 5 across all the terms in the trinomial. (2y)(3y 2 ) + (2y)(-8y) + (2y)(7) + (5)(3y 2 ) + (5)(-8y) + (5)(7) Group each term. (2)(3)(y)(y 2 ) + (2)(-8)(y)(y) + (2)(7)(y) + (5)(3)(y 2 ) + (5)(-8)(y) + (5)(7) Multiply each term out. 6y 3 + -16y 2 + 14y + 15y 2 + -40y + 35 Combine like terms and clean up the answer. 6y 3 - 16y 2 + 15y 2 + 14y - 40y + 35 6y 3 - 1y 2 - 26y + 35

6. © William James Calhoun, 2001 9-7: Multiplying Polynomials EXAMPLE 3: Find the product (x 2 + 4x - 5)(3x 2 - 7x + 2). Distribute the x 2, the 4x, and the -5 across all the terms in the second trinomial. (x 2 )(3x 2 ) + (x 2 )(-7x) + (x 2 )(2) + (4x)(3x 2 ) + (4x)(-7x) + (4x)(2) + (-5)(3x 2 ) + (-5)(-7x) + (-5)(2) Group each term. (3)(x 2 )(x 2 ) + (-7)(x 2 )(x) + (2)(x 2 ) + (4)(3)(x)(x 2 ) + (4)(-7)(x)(x) + (4)(2)(x) + (-5)(3)(x 2 ) + (-5)(-7)(x) + (-5)(2) Multiply each term out. 3x 4 + -7x 3 + 2x 2 + 12x 3 + -28x 2 + 8x + -15x 2 + 35x + -10 Combine like terms and clean up the answer. 3x 4 - 7x 3 + 12x 3 + 2x 2 - 28x 2 - 15x 2 + 8x + 35x - 10 3x 4 + 5x 3 - 41x 2 + 43x - 10

7. © William James Calhoun, 2001 9-7: Multiplying Polynomials EXAMPLE 4: Find the product (x 3 - 8x 2 + 9)(3x +4). Distribute the x 3, the -8x 2, and the 9 across the terms in the binomial. (x 3 )(3x) + (x 3 )(4) + (-8x 2 )(3x) + (-8x 2 )(4) + (9)(3x) + (9)(4) Group each term. (3)(x 3 )(x) + (4)(x 3 ) + (-8)(3)(x 2 )(x) + (-8)(4)(x 2 ) + (9)(3)(x) + (9)(4) Multiply each term out. 3x 4 + 4x 3 + -24x 3 + -32x 2 + 27x + 36 Combine like terms and clean up the answer. 3x 4 - 20x 3 - 32x 2 + 27x + 36

8. © William James Calhoun, 2001 9-7: Multiplying Polynomials HOMEWORK Page 539 #15 - 29 odd plus #’s 35 and 37