Concept explainers
For Exercises 43–46, use the remainder theorem to evaluate the polynomial for the given values of x. (See Example 6)
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- In Exercises 9–12, find a first- degree polynomial function P1 whose value and slope agree with the value and slope of f at x = c. Use a graphing utility to graph f and P1.arrow_forwardIn Exercises 16–17, find the zeros for each polynomial function and give the multiplicity of each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. 1)(x + 2)²(x + 5)³ 25x+125 16. f(x) = -2(x 17. f(x) = x³ - 5x²arrow_forwardIn Exercises 12–20, find all zeros of each polynomial function. Then graph the function. 12. f(x) = (x – 2)°(x + 1)³ 13. f(x) = -(x – 2)(x + 1)? 14. f(x) = x - xr? – 4x + 4 15. f(x) = x* - 5x² + 4 16. f(x) = -(x + 1)° 17. f(x) = -6x³ + 7x? - 1 18. f(x) = 2r³ – 2x 19. f(x) = x - 2x² + 26x 20. f(x) = -x + 5x² – 5x – 3 %3D %3D %3! %3D %3!arrow_forward
- In Exercises 26–31, find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. 26. n= 3; 4 and 2i are zeros; f(-1) = -50 31. n= 4; -2, 5, and 3 + 2i are zeros; f(1) = -96arrow_forwardIn Exercises 35–42, find all real values of x for which fx0. f(x)=4x+6arrow_forwardExercises 47 D–520: The graph of either a cubic, quartic, or quintic polynomial f(x) with integer zeros is shown. Write the complete factored form of f(x). (Hint: In Exercises 51 O and 52 O the leading coefficient is not +1.)arrow_forward
- In Exercises 1–16, divide using long division. State the quotient, q(x), and the remainder, r(x). 18x4 + 9x3 + 3x2 /3x2+1 In Exercises 17–25, divide using synthetic division. 17. (2x2 +x-10)/(x-2) 25. (x2 -5x-5x3 +x4)/(5+x)arrow_forwardIn Exercises 130–133, use a graphing utility to graph the functions y, and y2. Select a viewing rectangle that is large enough to show the end behavior of y2. What can you conclude? Verify your conclusions using polynomial multiplication. 130. yı = (x - 2)² y2 = x2 – 4x + 4 131. yı = (x – 4)(x² y2 = x - 7x2 + 14x – 8 132. yı = (x – 1)(x + x + 1) y2 = x – 1 133. yı = (x + 1.5)(x – 1.5) y2 = x? – 2.25 3x + 2)arrow_forwardIn Problems 17–28, determine which functions are polynomial functions. For those that are, state the degree. For those that are not, tellwhy not. Write each polynomial in standard form. Then identify the leading term and the constant termarrow_forward
- In Exercises 83–86, determine whether thestatement is true or false. If it is false, explain why or give anexample that shows it is false. If the graph of a polynomial function has three x-intercepts,then it must have at least two points at which its tangent line ishorizontal.arrow_forwardIn Exercises 9–16, a. List all possible rational zeros. b. Use synthetic division to test the possible rational zeros and find an actual zero. c. Use the quotient from part (b) to find the remaining zeros of the polynomial function. 9. f(x) = x + x² - 4x – 4 10. f(x) = x - 2x² – 11x + 12 11. f(x) = 2x - 3x? - 11x + 6 12. f(x) = 2r - 5x² + x + 2 13. f(x) = x + 4x² 14. f(x) = 2r + x² - 3x + 1 3x - 6 – 15. f(x) = 2r3 + 6x2 + 5x + 2 16. flx) = x - 4x² + &r – 5arrow_forwardIn Problems 99–106, analyze each polynomial function farrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage