1. Multiplying Polynomials and Special Products of Binomials 1-5 and 1-6 English Casbarro Unit 1 : Relations and Functions

2. 1-5: Multiplying Monomials  Remember! When multiplying powers with the same base, keep the base and add the exponents. x 2 · x 3 = x 2+3 = x 5  You will multiply the coefficients by each other just like you always have done. (5)(4) = 20  So, when you put the two items together, you have two monomials: (5x 2 )(4x 3 ) = 20x 5

3. Example 1: (–3x 3 y 2 )(4xy 5 ) = (–3 · 4)(x 3 · x)(y 2 · y 5 ) = (-12)(x 3 + 1 )(y 2 + 5 )= -12x 4 y 7 Example 2: Now you try: a. b. c.

5. Multiplying monomials and polynomials  Remember the distributive property when you are multiplying larger polynomials. The rules of the exponents are still true, but you have several terms that you now how to distribute.

6. Example 1: Example 2: You try: a.b.c.

7. Warm-up: Answer the following questions. 1. What is the degree of the monomial 5x y 4 z ? A. 6B. 1C. 4D. 5 2. For f(x) = 2x 2 + 4x – 6 and g(x) = 2x 2 + 2x - 8, find f(x) – g(x). A. – 4x 2 – 2x + 2 B. 2x + 2C. 4x 2 + 6x + 2D. 2x – 14 3. Which polynomial is written in standard form? A. 7 + 2x 4 – x 6 B. 3x 3 – x 5 C. x 4 D. x 2 + 3 – 2x 4. What is the degree of the polynomial function h(x) = 7x 3 – x 6 + x A. 10B. 3C. – 1D. 6 5. Short Response Evaluate for x = –2 P(x) and R(x) are polynomials. P(x) is a trinomial. Give examples of P(x) and R(x) that meet the following conditions. 6. P(x) – R(x) is a binomial. 7. P(x) – R(x) is a polynomial with 4 terms. 8. P(x) is a quartic.

8. Multiplying 2 binomials

9. You also can think of it as the double distributive– First, I have to multiply x by everything in the second parenthesis. Then I have to come through again and multiply 3 by everything in the second parenthesis. (x + 3)(x + 2) = x(x + 2) + 3(x + 2) = x(x) + x(2) + 3(x) + 3(2) = x 2 + 2x + 3x + 6 = x 2 + 5x + 6

10. Special products of binomials  You also need to be aware of 2 special types: 1) (a – b)(a + b)=a 2 – b 2 2) (a – b) 2 = a 2 – 2ab + b 2 (a + b) 2 = a 2 + 2ab + b 2 They multiply just like any other binomials, but later you will be factoring, and it will help you to remember these.

11. Example 1: (a – b)(a + b) = a(a) + a(b) – b(a) – b(b) = a 2 + ab – ab - b 2 = a 2 – b 2 Example 2: (a + b) 2 = ( a + b)(a + b) = a 2 + a(b) + b(a) – b(b) = a 2 + ab + ab - b 2 = a 2 + 2ab + b 2 Huge Note: Make sure that when you have a binomial squared, that you first write out the 2 binomials side by side.

12. Example 1:(x + 4) 2 = (x + 4)(x + 4) = x 2 + 4x + 4x + 16 = x 2 + 8x + 16 Example 2: (3x + 2y) 2 = (3x + 2y)(3x + 2y) = (3x)(3x) + (3x)(2y) + (3x)(2y) + (2y)(2y) = 9x 2 + 6xy + 6xy + 4y 2 = 9x 2 + 12xy + 4y 2 You try:a. (x – 5) 2 b. (6x -5) 2

13. Exponent Rules a m · a n = a m+n (a m ) n = a mn (ab) m =a m b m

14. Turn in the following problems. 1.Marie is planning a garden. She designs a rectangular garden with a length of (x + 4) and a width of (x + 1) feet. a. Draw a diagram of the garden and label it. b. Write a polynomial that represents the area of Marie’s garden. c. Find the area when x = 5. 2. Copy and complete the table below. A Degree of A B Degree of B A·B Degree of AB 2x22x2 23x53x5 56x76x7 7 5x35x3 2x 2 + 1 x 2 + 2x 2 – x x – 3x 3 – 2x 2 + 1 a. b. c. d. Use the results of the chart to complete the following: The product of a polynomial of degree m and a polynomial of degree n, has a degree of_______.