Dividing Polynomials

1. If division of a polynomial by a binomial results in a remainder of zero, what can be conclude?

For the following exercises, use long division to divide. Specify the quotient and the remainder.

3. [latex]\left({x}^{2}+5x - 1\right)\div \left(x - 1\right)[/latex]

7. [latex]\left(6{x}^{2}-25x - 25\right)\div \left(6x+5\right)[/latex]

13. [latex]\left(2{x}^{3}+3{x}^{2}-4x+15\right)\div \left(x+3\right)[/latex]

For the following exercises, use synthetic division to find the quotient.

15. [latex]\left(2{x}^{3}-6{x}^{2}-7x+6\right)\div \left(x - 4\right)[/latex]

19. [latex]\left(3{x}^{3}-2{x}^{2}+x - 4\right)\div \left(x+3\right)[/latex]

23. [latex]\left(4{x}^{3}-5{x}^{2}+13\right)\div \left(x+4\right)[/latex]

29. [latex]\left({x}^{4}+{x}^{3}-3{x}^{2}-2x+1\right)\div \left(x+1\right)[/latex]

For the following exercises, use synthetic division to find the quotient and remainder.

43. [latex]\frac{4{x}^{3}-33}{x - 2}[/latex]

45. [latex]\frac{3{x}^{3}+2x - 5}{x - 1}[/latex]

For the following exercises, use the given volume of a box and its length and width to express the height of the box algebraically.

61. Volume is [latex]12{x}^{3}+20{x}^{2}-21x - 36[/latex], length is [latex]2x+3[/latex], width is [latex]3x - 4[/latex].

63. Volume is [latex]10{x}^{3}+27{x}^{2}+2x - 24[/latex], length is [latex]5x - 4[/latex], width is [latex]2x+3[/latex].

Complex Numbers

1. Explain how to add complex numbers.

3. Give an example to show the product of two imaginary numbers is not always imaginary.

For the following exercises, evaluate the algebraic expressions.

7. [latex]\text{If }f\left(x\right)={x}^{2}+3x+5[/latex], evaluate [latex]f\left(2+i\right)[/latex].

9. [latex]\text{If }f\left(x\right)=\frac{x+1}{2-x}[/latex], evaluate [latex]f\left(5i\right)[/latex].

For the following exercises, plot the complex numbers on the complex plane.

13. [latex]1 - 2i[/latex]

15. i

For the following exercises, perform the indicated operation and express the result as a simplified complex number.

17. [latex]\left(3+2i\right)+\left(5 - 3i\right)[/latex]

19. [latex]\left(-5+3i\right)-\left(6-i\right)[/latex]

23. [latex]\left(5 - 2i\right)\left(3i\right)[/latex]

25. [latex]\left(-2+4i\right)\left(8\right)[/latex]

27. [latex]\left(-1+2i\right)\left(-2+3i\right)[/latex]

29. [latex]\left(3+4i\right)\left(3 - 4i\right)[/latex]

33. [latex]\frac{6+4i}{i}[/latex]

35. [latex]\frac{3+4i}{2-i}[/latex]

37. [latex]\sqrt{-9}+3\sqrt{-16}[/latex]

39. [latex]\frac{2+\sqrt{-12}}{2}[/latex]

41. [latex]{i}^{8}[/latex]

43. [latex]{i}^{22}[/latex]

Zeros of Polynomial Functions

3. What is the difference between rational and real zeros?

5. If synthetic division reveals a zero, why should we try that value again as a possible solution?

For the following exercises, use the Remainder Theorem to find the remainder.

7. [latex]\left(3{x}^{3}-2{x}^{2}+x - 4\right)\div \left(x+3\right)[/latex]

11. [latex]\left({x}^{4}-1\right)\div \left(x - 4\right)[/latex]

13. [latex]\left(4{x}^{3}+5{x}^{2}-2x+7\right)\div \left(x+2\right)[/latex]

For the following exercises, use the Factor Theorem to find all real zeros for the given polynomial function and one factor.

15. [latex]f\left(x\right)=2{x}^{3}+{x}^{2}-5x+2;\text{ }x+2[/latex]

21. [latex]2{x}^{3}+5{x}^{2}-12x - 30,\text{ }2x+5[/latex]

For the following exercises, use the Rational Zero Theorem to find all real zeros.

23. [latex]2{x}^{3}+7{x}^{2}-10x - 24=0[/latex]

25. [latex]{x}^{3}+5{x}^{2}-16x - 80=0[/latex]

29. [latex]2{x}^{3}-3{x}^{2}-x+1=0[/latex]

31. [latex]2{x}^{3}-5{x}^{2}+9x - 9=0[/latex]

33. [latex]{x}^{4}-2{x}^{3}-7{x}^{2}+8x+12=0[/latex]

37. [latex]{x}^{4}+2{x}^{3}-4{x}^{2}-10x - 5=0[/latex]

For the following exercises, find all complex solutions (real and non-real).

41. [latex]{x}^{3}-8{x}^{2}+25x - 26=0[/latex]

45. [latex]2{x}^{3}-3{x}^{2}+32x+17=0[/latex]

For the following exercises, list all possible rational zeros for the functions.

57. [latex]f\left(x\right)=2{x}^{{}^{3}}+3{x}^{2}-8x+5[/latex]

59. [latex]f\left(x\right)=6{x}^{4}-10{x}^{2}+13x+1[/latex]

For the following exercises, use your calculator to graph the polynomial function. Based on the graph, find the rational zeros. All real solutions are rational.

61. [latex]f\left(x\right)=6{x}^{3}-7{x}^{2}+1[/latex]

63. [latex]f\left(x\right)=8{x}^{3}-6{x}^{2}-23x+6[/latex]

65. [latex]f\left(x\right)=16{x}^{4}-24{x}^{3}+{x}^{2}-15x+25[/latex]

For the following exercises, find the dimensions of the box described.

71. The length is twice as long as the width. The height is 2 inches greater than the width. The volume is 192 cubic inches.

75. The length is 3 inches more than the width. The width is 2 inches more than the height. The volume is 120 cubic inches.

Inverse and Radical Functions

1. Explain why we cannot find inverse functions for all polynomial functions.

3. When finding the inverse of a radical function, what restriction will we need to make?

For the following exercises, find the inverse of the function on the given domain.

5. [latex]f\left(x\right)={\left(x - 4\right)}^{2}, \left[4,\infty \right)[/latex]

7. [latex]f\left(x\right)={\left(x+1\right)}^{2}-3, \left[-1,\infty \right)[/latex]

9. [latex]f\left(x\right)=3{x}^{2}+5,\left(-\infty ,0\right],\left[0,\infty \right)[/latex]

11. [latex]f\left(x\right)=9-{x}^{2}, \left[0,\infty \right)[/latex]

31. [latex]f\left(x\right)={x}^{2}-6x+3, \left[3,\infty \right)[/latex]

For the following exercises, find the inverse of the functions.

13. [latex]f\left(x\right)={x}^{3}+5[/latex]

17. [latex]f\left(x\right)=\sqrt{2x+1}[/latex]

21. [latex]f\left(x\right)=9+2\sqrt[3]{x}[/latex]

23. [latex]f\left(x\right)=\frac{2}{x+8}[/latex]

25. [latex]f\left(x\right)=\frac{x+3}{x+7}[/latex]

For the following exercises, find the inverse of the function and graph both the function and its inverse.

35. [latex]f\left(x\right)={\left(x - 4\right)}^{2},x\ge 4[/latex]

37. [latex]f\left(x\right)=1-{x}^{3}[/latex]

41. [latex]f\left(x\right)=\frac{1}{{x}^{2}},x\ge 0[/latex]

For the following exercises, determine the function described and then use it to answer the question.

57. An object dropped from a height of 200 meters has a height, [latex]h\left(t\right)[/latex], in meters after t seconds have lapsed, such that [latex]h\left(t\right)=200 - 4.9{t}^{2}[/latex]. Express t as a function of height, h, and find the time to reach a height of 50 meters.

59. The volume, V, of a sphere in terms of its radius, r, is given by [latex]V\left(r\right)=\frac{4}{3}\pi {r}^{3}[/latex]. Express r as a function of V, and find the radius of a sphere with volume of 200 cubic feet.