Exponents and Polynomials - XYZ Custom Plus

582Chapter 9 Exponents and Polynomials5. Multiply: (x + 2)(x + 6)Example 5Multiply: (x + 3)(x + 5)Solution We can think of the first binomial, x + 3, as a single number.(Remember, for any value of x, x + 3 will be just a number.) We apply the distributiveproperty by multiplying x + 3 times both x and 5.(x + 3)(x + 5) = (x + 3) ⋅ x + (x + 3) ⋅ 5Next, we apply the distributive property again to multiply x times both x and 3,and 5 times both x and 3.= x ⋅ x + 3 ⋅ x + x ⋅ 5 + 3 ⋅ 5= x 2 + 3x + 5x + 15The last thing to do is to combine the similar terms 3x and 5x to get 8x.(Remember, this is also an application of the distributive property.)= x 2 + 8x + 156. Multiply: (x − 2)(x + 6)Example 6Multiply: (x − 3)(x + 5)Solution The only difference between the binomials in this example and thosein Example 5 is the subtraction sign in x − 3. The steps in multiplying are exactlythe same.(x − 3)(x + 5) = (x − 3) ⋅ x + (x − 3) ⋅ 5 Multiply x − 3 timesboth x and 5= x ⋅ x − 3 ⋅ x + x ⋅ 5 − 3 ⋅ 5 Distributive propertytwo more times= x 2 − 3x + 5x − 15 Simplify each term= x 2 + 2x − 15 −3x − 5x − 2x7. Multiply: (3x − 2)(5x + 4)Example 7Multiply: (2x − 3)(4x + 7)SolutionUsing the same steps shown in Examples 5 and 6, we have( 2 x − 3)(4x + 7) = (2x − 3) ⋅ 4x + (2x − 3) ⋅ 7= 2x ⋅ 4x − 3 ⋅ 4x + 2x ⋅ 7 − 3 ⋅ 7= 8x 2 − 12x + 14x − 21= 8x 2 + 2x − 21Our next two examples show how we raise binomials to the second power.8. Expand and multiply: (x + 3) 2Example 8Expand and multiply: (x + 5) 2Solution We use the definition of exponents to write (x + 5) 2 as(x + 5)(x + 5). Then we multiply as we did in the previous examples.(x + 5) 2 = (x + 5)(x + 5) Definition of exponents= (x + 5) ⋅ x + (x + 5) ⋅ 5 Distributive property= x ⋅ x + 5 ⋅ x + x ⋅ 5 + 5 ⋅ 5 Distributive property= x 2 + 5x + 5x + 25 Simplify each term= x 2 + 10x + 25 5x + 5x = 10xAnswers5. x 2 + 8x + 126. x 2 + 4x − 12 7. 15x 2 + 2x − 88. x 2 + 6x + 9