b) The GCF of the terms in the equation is 4w. Remember, however, that we can divide an equation by a constant but we cannot divide an equation by a variable. Dividing by a variable may eliminate a solution and may mean we are dividing by zero. So let’s begin by dividing each term by 4. w3 4 4 100w 4 0 4 Divide by 4. w3 25w 0 Simplify. w(w2 25) 0 Factor out w. w(w 5)(w 5) 0 Factor w2 25. w 0 or w 5 0 or w 5 0 Set each factor equal to zero. w 5 w 5 Solve. Check. The solution set is {0, 5, 5}. YOU TRY 4 Solve. a) (c 10)(2c2 5c 7) 0 b) r4 25r2 In this section, it was possible to solve all of the equations by factoring. Below we show the relationship between solving a quadratic equation by factoring and solving it using a graphing calculator. In Chapter 10 we will learn other methods for solving quadratic equations. Using Technology In this section, we learned how to solve a quadratic equation by factoring. We can use a graphing calculator to solve a quadratic equation as well. Let’s see how the two are related by using the equation x2 x 6 0. x2 x 6 0 (x 2)(x 3) 0 x 2 0 or x 3 0 x 2 x 3 The solution set is {2, 3}. Next, solve x2 x 6 0 using a graphing calculator. Recall from Chapter 4 that to fi nd the x-intercepts of the graph of an equation we let y 0 and solve the equation for x. If we let y x2 x 6, then solving x2 x 6 0 is the same as fi nding the x-intercepts of the graph of y x2 x 6. X-intercepts are also called zeros of the equation since they are the values of x that make y 0. Use Y to enter y x2 x 6 into the calculator, press ZOOM , and then enter 6 to display the graph using the standard viewing window as shown at right. We obtain a graph called a parabola, and we can see that it has two x-intercepts. If the scale for each tick mark on the graph is 1, then it appears that the x-intercepts are 2 and 3. To verify press TRACE, type 2, and press ENTER. Since x 2 and y 0, x 2 is an x-intercept. While still in “Trace mode,” type 3 and press ENTER. Since x 3 and y 0, x 3 is an x-intercept. 398 CHAPTER 7 Factoring Polynomials www.mhhe.com/messersmith
messersmith_power_intermediate_algebra_1e_ch4_7_10
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