Example 2 3x 42x 5 3x2x 3x5 42x 45 6x 2 15x 8x 20 6x 2 7x 20 To verify let x 2

Example 2 3x 42x 5 3x2x 3x5 42x 45 6x 2 15x 8x 20 6x

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Example 2 (3x 4)(2x + 5) = (3x)(2x) + (3x)(5) + (-4)(2x) + (-4)(5) = 6x 2 + 15x 8x 20 = 6x 2 + 7x 20 To verify, let x = 2 (3x 4)(2x + 5) = 6x 2 + 7x 20 (3(2) 4)(2(2) + 5) 6(2) 2 + 7(2) 20 (6 4)(4 + 5) 6(4) + 14 20 (2)(9) 24 6 18 18 Since both sides equal 18, (3x 4)(2x + 5) = 6x 2 + 7x 20 The algebraic method is most widely used and will be the most efficient method when determining more complex products such as the one that follows below. N.B . We can use any value in our check! N.B. Notice the format of a verification! There is a ? over the equal sign and a line separating the two sides of the equation. This is necessary until we have shown the two sides to be equal! ? ? Notice order of operations! We square 2 then multiply by 6.
10C P1-2 DETAILED SOLUTIONS Page 28 of 47 Example (3x 2 + 2)(x 2 2x) = (3x 2 )(x 2 ) + (3x 2 )(-2x) + (2)(x 2 ) + (2)(-2x) = 3x 4 6x 3 + 2x 2 4x Actually, we can use algebra and the distributive property to multiply any number of terms by any number of terms. Let‟s show this by multiplying a trinomial times a trinomial as in the example that follows. Example (x 2 + 2x + 3)(2x 2 3x + 1) First term in first trinomial is multiplied by each term in second trinomial = (x 2 )(2x 2 ) + (x 2 )(-3x) + (x 2 )(1) = 2x 4 3x 3 + x 2 result 1 Second term in first trinomial is multiplied by each term in second trinomial = (2x)(2x 2 ) + (2x)(-3x) + (2x)(1) = 4x 3 6x 2 + 2x result 2 Third term in first trinomial is multiplied by each term in second trinomial = (3)(2x 2 ) + (3)(-3x) + (3)(1) = 6x 2 9x + 3 result 3 The final product is result 1 + result 2 + result 3 . 2x 4 3x 3 + x 2 + 4x 3 6x 2 + 2x + 6x 2 9x + 3 = 2x 4 3x 3 + 4x 3 + x 2 6x 2 + 6x 2 + 2x 9x + 3 = 2x 4 + x 3 + x 2 7x + 3 When multiplying polynomials, it is important to remember that each term in the first polynomial is multiplied by each term in the second polynomial. Simplify the expression by combining like terms.
10C P1-2 DETAILED SOLUTIONS Page 29 of 47 Let‟s verify using x = -1 It is necessary to place -1 in parentheses as shown below! (x 2 + 2x + 3)(2x 2 3x + 1) = 2x 4 + x 3 + x 2 7x + 3 ((-1) 2 + 2(-1) + 3)(2(-1) 2 3(-1) + 1) 2(-1) 4 + (-1) 3 + (-1) 2 7(-1) + 3 (1 + -2 + 3)(2 + 3 + 1) 2 + -1 + 1 + 7 + 3 (2)(6) 13 + -1 12 12 Since both sides worked out to be 12, (x 2 + 2x + 3)(2x 2 3x + 1) = 2x 4 + x 3 + x 2 7x + 3 Now that you understand how to multiply polynomials, you will be asked to demonstrate your understanding in a variety of ways. Carefully read the following examples. Example 1 Suzie‟s solution to a multiplication question appears below. Identify and explain the error that she made. (3x 2)(2x + 5) = (3x)(2x) + (-2)(5) = 6x 2 10 Example 2 Explain to Justin the error that he made when he multiplied the following binomials. (4x 1)(2x 3) = (4x)(2x) + (4x)(-3) = 8x 2 12x Refer to Foundations and Pre-Calculus Mathematics 10 . On page 167, complete Question 13 To summarize, when multiplying polynomials it is necessary to Multiply each term in the first polynomial by each term in the second polynomial.

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