3.3E: Graphs of Polynomial Functions (Exercises)

Find the C and t intercepts of each function. \[1. C\left(t\right)=2\left(t-4\right)\left(t+1\right)(t-6) 2. C\left(t\right)=3\left(t+2\right)\left(t-3\right)(t+5)\] \[3. C\left(t\right)=4t\left(t-2\right)^{2} (t+1) 4. C\left(t\right)=2t\left(t-3\right)\left(t+1\right)^{2}\] \[5. C\left(t\right)=2t^{4} -8t^{3} +6t^{2} 6. C\left(t\right)=4t^{4} +12t^{3} -40t^{2}\]

Use your calculator or other graphing technology to solve graphically for the zeros of the function. \[7. f\left(x\right)=x^{3} -7x^{2} +4x+30 8. g\left(x\right)=x^{3} -6x^{2} +x+28\]

Find the long run behavior of each function as \(t\to \infty\) and \(t\to -\infty\) \[9. h\left(t\right)=3\left(t-5\right)^{3} \left(t-3\right)^{3} (t-2) 10. k\left(t\right)=2\left(t-3\right)^{2} \left(t+1\right)^{3} (t+2)\] \[11. p\left(t\right)=-2t\left(t-1\right)\left(3-t\right)^{2} 12. q\left(t\right)=-4t\left(2-t\right)\left(t+1\right)^{3}\]

Sketch a graph of each equation. \[13. f\left(x\right)=\left(x+3\right)^{2} (x-2) 14. g\left(x\right)=\left(x+4\right)\left(x-1\right)^{2}\] \[15. h\left(x\right)=\left(x-1\right)^{3} \left(x+3\right)^{2} 16. k\left(x\right)=\left(x-3\right)^{3} \left(x-2\right)^{2}\] \[17. m\left(x\right)=-2x\left(x-1\right)(x+3) 18. n\left(x\right)=-3x\left(x+2\right)(x-4)\]

Solve each inequality. \[19. \left(x-3\right)\left(x-2\right)^{2} >0 20. \left(x-5\right)\left(x+1\right)^{2} >0\] \[21. \left(x-1\right)\left(x+2\right)\left(x-3\right)<0 22. \left(x-4\right)\left(x+3\right)\left(x+6\right)<0\]

Find the domain of each function. \[23. f\left(x\right)=\sqrt{-42+19x-2x^{2} } 24. g\left(x\right)=\sqrt{28-17x-3x^{2} }\] \[25. h\left(x\right)=\sqrt{4-5x+x^{2} } 26. k\left(x\right)=\sqrt{2+7x+3x^{2} }\] \[27. n\left(x\right)=\sqrt{\left(x-3\right)\left(x+2\right)^{2} } 28. m\left(x\right)=\sqrt{\left(x-1\right)^{2} (x+3)}\] \[29. p\left(t\right)=\frac{1}{t^{2} +2t-8} 30. q\left(t\right)=\frac{4}{x^{2} -4x-5}\] Write an equation for a polynomial the given features.

1. Degree 3. Zeros at x = -2, x = 1, and x = 3. Vertical intercept at (0, -4)

2. Degree 3. Zeros at x = -5, x = -2, and x = 1. Vertical intercept at (0, 6)

3. Degree 5. Roots of multiplicity 2 at x = 3 and x = 1, and a root of multiplicity 1 at x = -3. Vertical intercept at (0, 9)

4. Degree 4. Root of multiplicity 2 at x = 4, and a roots of multiplicity 1 at x = 1 and x = -2. Vertical intercept at (0, -3)

5. Degree 5. Double zero at x = 1, and triple zero at x = 3. Passes through the point (2, 15)

6. Degree 5. Single zero at x = -2 and x = 3, and triple zero at x = 1. Passes through the point (2, 4)

Write a formula for each polynomial function graphed.

37. 38. 39.

40. 41. 42.

43. 44.

Write a formula for each polynomial function graphed.

45. 46.

47. 48.

49. 50.

1. A rectangle is inscribed with its base on the x axis and its upper corners on the parabola \(y=5-x^{2}\). What are the dimensions of such a rectangle that has the greatest possible area?

A rectangle is inscribed with its base on the x axis and its upper corners on the curve \(y=16-x^{4}\). What are the dimensions of such a rectangle that has the greatest possible area?203

3.4 Factor Theorem and Remainder Theorem