Power and Polynomial Functions: Get Stronger Answer Key

Introduction to Power and Polynomial Functions

  1. Power function
  2. Neither
  3. Neither
  4. Degree = [latex]2[/latex], Coefficient = [latex]-2[/latex]
  5. Degree = [latex]4[/latex], Coefficient =[latex]-2[/latex]
  6. As [latex]x\to\infty, f(x)\to\infty[/latex], as [latex]x\to-\infty, f(x)\to\infty[/latex]
  7. As [latex]x\to-\infty, f(x)\to-\infty[/latex], as [latex]x\to\infty, f(x)\to-\infty[/latex]
  8. As [latex]x\to-\infty, f(x)\to-\infty[/latex], as [latex]x\to\infty, f(x)\to-\infty[/latex]
  9. As [latex]x\to\infty, f(x)\to\infty[/latex], as [latex]x\to-\infty, f(x)\to-\infty[/latex]
  10. [latex]y[/latex]-intercept is [latex](0,12)[/latex], [latex]x[/latex]-intercepts are [latex](1,0)[/latex]; [latex](-2,0)[/latex]; and [latex](3,0)[/latex].
  11. [latex]y[/latex]-intercept is [latex](0,-16)[/latex], [latex]x[/latex]-intercepts are [latex](2,0)[/latex] and [latex](-2,0)[/latex].
  12. [latex]y[/latex]-intercept is [latex](0,0)[/latex], [latex]x[/latex]-intercepts are [latex](0,0)[/latex], ([latex]4,0)[/latex], and [latex](-2,0)[/latex].
  13. [latex]3[/latex]
  14. [latex]5[/latex]
  15. [latex]3[/latex]
  16. [latex]5[/latex]
  17. Yes. Number of turning points is [latex]2[/latex]. Least possible degree is [latex]3[/latex].
  18. Yes. Number of turning points is [latex]1[/latex]. Least possible degree is [latex]2[/latex].
  19. Yes. Number of turning points is [latex]0[/latex]. Least possible degree is [latex]1[/latex].
  20. Yes. Number of turning points is [latex]0[/latex]. Least possible degree is [latex]1[/latex].
  21. [latex]V(m) = 8m^3 + 36m^2 + 54m + 27[/latex]
  22. [latex]V(x) = 4x^3 - 32x^2 + 64x[/latex]

Graphs of Polynomial Functions

  1. [latex](-2,0), (3,0), (-5,0)[/latex]
  2. [latex](3,0), (-1,0), (0,0)[/latex]
  3. [latex](0,0), (-5,0), (2,0)[/latex]
  4. [latex](0,0), (-5,0), (4,0)[/latex]
  5. [latex](2,0), (-2,0), (-1,0)[/latex]
  6. [latex](-2,0), (2,0), (1/2,0)[/latex]
  7. [latex](1,0), (-1,0)[/latex]
  8. [latex](0,0), (\sqrt{3},0), (-\sqrt{3},0)[/latex]
  9. [latex](0,0), (1,0), (-1,0), (2,0), (-2,0)[/latex]
  10. [latex]f(2)=-10[/latex] and [latex]f(4)=28[/latex]. Sign change confirms.
  11. [latex]f(1)=3[/latex] and [latex]f(3)=-77[/latex]. Sign change confirms.
  12. [latex]f(0.01)=1.000001[/latex] and [latex]f(0.1)=-7.999[/latex]. Sign change confirms.
  13. [latex]0[/latex] with multiplicity [latex]2[/latex], [latex]-\dfrac{3}{2}[/latex] with multiplicity [latex]5[/latex], [latex]4[/latex] with multiplicity [latex]2[/latex]
  14. [latex]0[/latex] with multiplicity [latex]2[/latex], [latex]-2[/latex] with multiplicity [latex]2[/latex]
  15. [latex]-\dfrac{2}{3}[/latex] with multiplicity [latex]5[/latex], [latex]5[/latex] with multiplicity [latex]2[/latex]
  16. [latex]0[/latex] with multiplicity [latex]4[/latex], [latex]2[/latex] with multiplicity [latex]1[/latex], [latex]-1[/latex] with multiplicity [latex]1[/latex]
  17. [latex]\dfrac{3}{2}[/latex] with multiplicity [latex]2[/latex], [latex]0[/latex] with multiplicity [latex]3[/latex]
  18. [latex]0[/latex] with multiplicity [latex]6[/latex], [latex]\dfrac{2}{3}[/latex] with multiplicity [latex]2[/latex]
  19. [latex]x[/latex]-intercepts, [latex](1,0)[/latex] with multiplicity [latex]2[/latex], [latex](-4,0)[/latex] with multiplicity [latex]1[/latex], [latex]y[/latex]-intercept [latex](0,4)[/latex]. As [latex]x\to-\infty, f(x)\to-\infty[/latex], as [latex]x\to\infty, f(x)\to\infty[/latex].
    Graph of g(x)=(x+4)(x-1)^2.
  20. [latex]x[/latex]-intercepts [latex](3,0)[/latex] with multiplicity [latex]3[/latex], [latex](2,0)[/latex] with multiplicity [latex]2[/latex], [latex]y[/latex]-intercept [latex](0,-108)[/latex]. As [latex]x\to-\infty, f(x)\to-\infty[/latex], as [latex]x\to\infty, f(x)\to\infty[/latex].
    Graph of k(x)=(x-3)^3(x-2)^2.
  21. [latex]x[/latex]-intercepts [latex](0,0)[/latex], [latex](-2,0)[/latex], [latex](4,0)[/latex] with multiplicity [latex]1[/latex], [latex]y[/latex]-intercept [latex](0,0)[/latex]. As [latex]x\to-\infty, f(x)\to\infty[/latex], as [latex]x\to\infty, f(x)\to-\infty[/latex].
    Graph of n(x)=-3x(x+2)(x-4).
  22. [latex]f(x)=-\dfrac{2}{5}(x-3)(x+1)(x+3)[/latex]
  23. [latex]f(x)=\dfrac{1}{4}(x+2)^2(x-3)[/latex]
  24. [latex]-4[/latex], [latex]-2[/latex], [latex]1[/latex], [latex]3[/latex] with multiplicity [latex]1[/latex]
  25. [latex]-2[/latex], [latex]3[/latex] each with multiplicity [latex]2[/latex]
  26. [latex]f(x)=-\dfrac{2}{3}(x+2)(x-1)(x-3)[/latex]
  27. [latex]f(x)=\dfrac{1}{3}(x-3)^2(x-1)^2(x+3)[/latex]
  28. [latex]f(x)=-15(x-1)^2(x-3)^3[/latex]
  29. [latex]f(x)=-2(x+3)(x+2)(x-1)[/latex]
  30. [latex]f(x)=-\dfrac{3}{2}(2x-1)^2(x-6)(x+2)[/latex]

Dividing Polynomials

  1. [latex]x+6+\dfrac{5}{x-1}[/latex], quotient: [latex]x+6[/latex], remainder: [latex]5[/latex]
  2. [latex]3x+2[/latex], quotient: [latex]3x+2[/latex], remainder: [latex]0[/latex]
  3. [latex]x-5[/latex], quotient: [latex]x-5[/latex], remainder: [latex]0[/latex]
  4. [latex]2x-7+\dfrac{16}{x-2}[/latex], quotient: [latex]2x-7[/latex], remainder: [latex]16[/latex]
  5. [latex]2x^2+2x+1+\dfrac{10}{x-4}[/latex]
  6. [latex]2x^2-7x+1-\dfrac{2}{2x+1}[/latex]
  7. [latex]3x^2-11x+34-\dfrac{106}{x+3}[/latex]
  8. [latex]x^2+5x+1[/latex]
  9. [latex]4x^2-21x+84-\dfrac{323}{x+4}[/latex]
  10. [latex]x^2-14x+49[/latex]
  11. Yes [latex](x-2)(3x^2-5)[/latex]
  12. Yes [latex](x-2)(4x^3+8x^2+x+2)[/latex]
  13. No

Zeros of Polynomial Functions

  1. [latex]-106[/latex]
  2. [latex]0[/latex]
  3. [latex]255[/latex]
  4. [latex]-1[/latex]
  5. [latex]-2[/latex], [latex]1[/latex], [latex]\dfrac{1}{2}[/latex]
  6. [latex]-2[/latex]
  7. [latex]-3[/latex]
  8. [latex]-\dfrac{5}{2}[/latex], [latex]\sqrt{6}[/latex], [latex]-\sqrt{6}[/latex]
  9. [latex]2[/latex], [latex]-4[/latex], [latex]-\dfrac{3}{2}[/latex]
  10. [latex]4[/latex], [latex]-4[/latex], [latex]-5[/latex]
  11. [latex]5[/latex], [latex]-3[/latex], [latex]-\dfrac{1}{2}[/latex]
  12. [latex]\dfrac{1}{2}[/latex], [latex]\dfrac{1+\sqrt{5}}{2}[/latex], [latex]\dfrac{1-\sqrt{5}}{2}[/latex]
  13. [latex]\dfrac{3}{2}[/latex]
  14. [latex]2[/latex], [latex]3+2i[/latex], [latex]3-2i[/latex]
  15. [latex]-\dfrac{2}{3}[/latex], [latex]1+2i[/latex], [latex]1-2i[/latex]
  16. [latex]-\dfrac{1}{2}[/latex], [latex]1+4i[/latex], [latex]1-4i[/latex]
  17. 1 positive, 1 negative
    Graph of f(x)=x^4-x^2-1.
  18. 3 or 1 positive, 0 negative
    Graph of f(x)=x^3-2x^2+x-1.
  19. 0 positive, 3 or 1 negative
    Graph of f(x)=2x^3+37x^2+200x+300.
  20. 2 or 0 positive, 2 or 0 negative
    Graph of f(x)=2x^4-5x^3-5x^2+5x+3.
  21. 2 or 0 positive, 2 or 0 negative
    Graph of f(x)=10x^4-21x^2+11.
  22. [latex]\pm5[/latex], [latex]\pm1[/latex], [latex]\pm\dfrac{5}{2}[/latex], [latex]\pm\dfrac{1}{2}[/latex]
  23. [latex]\pm1[/latex], [latex]\pm\dfrac{1}{2}[/latex], [latex]\pm\dfrac{1}{3}[/latex], [latex]\pm\dfrac{1}{6}[/latex]
  24. [latex]8[/latex] by [latex]4[/latex] by [latex]6[/latex] inches
  25. [latex]5.5[/latex] by [latex]4.5[/latex] by [latex]3.5[/latex] inches
  26. [latex]8[/latex] by [latex]5[/latex] by [latex]3[/latex] inches
  27. Radius = [latex]6[/latex] meters, Height = [latex]2[/latex] meters
  28. Radius = [latex]2.5[/latex] meters, Height = [latex]4.5[/latex] meters