340 CHAPTER 3 Polynomial and Rational Functions 86. A quadratic equation ƒ1x2 = 0 has a solution x = 2. Its graph has vertex 15, 32. What is the other solution of the equation? Relating Concepts For individual or collaborative investigation (Exercises 87–90) A quadratic inequality such as x2 + 2x - 8 60 can be solved by first solving the related quadratic equation x2 + 2x - 8 = 0, identifying intervals determined by the solutions of this equation, and then using a test value from each interval to determine which intervals form the solution set. Work Exercises 87–90 in order to learn a graphical method of solving inequalities. 87. Graph ƒ1x2 = x2 + 2x - 8. 88. The real solutions of x2 + 2x - 8 = 0 are the x-values of the x-intercepts of the graph in Exercise 87. These are values of x for which ƒ1x2 = 0. What are these values? What is the solution set of this equation? 89. The real solutions of x2 + 2x - 8 60 are the x-values for which the graph in Exercise 87 lies below the x-axis. These are values of x for which ƒ1x2 60 is true. What interval of x-values represents the solution set of this inequality? 90. The real solutions of x2 + 2x - 8 70 are the x-values for which the graph in Exercise 87 lies above the x-axis. These are values of x for which ƒ1x2 70 is true. What intervals of x-values represent the solution set of this inequality? Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. 91. x2 - x - 6 60 92. x2 - 9x + 20 60 93. 2x2 - 9x Ú 18 94. 3x2 + x Ú 4 95. -x2 + 4x + 1 Ú 0 96. -x2 + 2x + 6 70 3.2 Synthetic Division ■ Synthetic Division ■ Remainder Theorem ■ Potential Zeros of Polynomial Functions The outcome of a division problem can be written using multiplication, even when the division involves polynomials. The division algorithm illustrates this. Division Algorithm Let ƒ1x2 and g1x2 be polynomials with g1x2 of lesser degree than ƒ1x2 and g1x2 of degree 1 or more. There exist unique polynomials q1x2 and r1x2 such that ƒ1x2 =g1x2 # q1x2 +r1x2, where either r1x2 = 0 or the degree of r1x2 is less than the degree of g1x2.
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