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

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

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

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.

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

Warm-up: Answer the following questions. 1. What is the degree of the monomial 5x y 4 z ? A. 6B. 1C. 4D 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 – Which polynomial is written in standard form? A x 4 – x 6 B. 3x 3 – x 5 C. x 4 D. x – 2x 4. What is the degree of the polynomial function h(x) = 7x 3 – x 6 + x A. 10B. 3C. – 1D 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.

Multiplying 2 binomials

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

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.

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.

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 xy + 4y 2 You try:a. (x – 5) 2 b. (6x -5) 2

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

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 = 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 x 2 + 2x 2 – x x – 3x 3 – 2x 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_______.