Polynomial Equations: Get Stronger Key — Precalculus Practice Page Answer Keys

Dividing Polynomials Solutions

1. The binomial is a factor of the polynomial.

2. [latex]x+6+\frac{5}{x - 1}\text{,}\text{quotient:}x+6\text{,}\text{remainder:}\text{5}[/latex]

3. [latex]x - 5\text{,}\text{quotient: }x - 5\text{,}\text{remainder: }\text{0}[/latex]

4. [latex]2{x}^{2}-3x+5\text{,}\text{quotient:}2{x}^{2}-3x+5\text{,}\text{remainder: }\text{0}[/latex]

5. [latex]2{x}^{2}+2x+1+\frac{10}{x - 4}[/latex]

6. [latex]3{x}^{2}-11x+34-\frac{106}{x+3}[/latex]

7. [latex]4{x}^{2}-21x+84-\frac{323}{x+4}[/latex]

8. [latex]{x}^{3}-3x+1[/latex]

9. [latex]\text{Quotient: }4{x}^{2}+8x+16\text{,}\text{remainder: }-1[/latex]

10. [latex]\text{Quotient: }3{x}^{2}+3x+5\text{,}\text{remainder: }0[/latex]

11. [latex]2x+3[/latex]

12. [latex]x+2[/latex]

Complex Numbers Solutions

1. Add the real parts together and the imaginary parts together.

2. i times i equals –1, which is not imaginary. (answers vary)

3. [latex]14+7i[/latex]

4. [latex]-\frac{23}{29}+\frac{15}{29}i[/latex]

5.
Graph of the plotted point, 1-2i.

6.
Graph of the plotted point, i.

7. [latex]8-i[/latex]

8. [latex]-11+4i[/latex]

9. [latex]6+15i[/latex]

10. [latex]-16+32i[/latex]

11. [latex]-4 - 7i[/latex]

12. 25

13. [latex]4 - 6i[/latex]

14. [latex]\frac{2}{5}+\frac{11}{5}i[/latex]

15. 15i

16. [latex]1+i\sqrt{3}[/latex]

17. 1

18. –1

Zeros of Polynomial Functions Solutions

1. Rational zeros can be expressed as fractions whereas real zeros include irrational numbers.

2. Polynomial functions can have repeated zeros, so the fact that number is a zero doesn’t preclude it being a zero again.

3. –106

[unmatched: no problem 9 shown in this exercise set — key answer is “0”]

4. 255

5. –1

6. –2, 1, [latex]\frac{1}{2}[/latex]

7. [latex]-\frac{5}{2}, \sqrt{6}, -\sqrt{6}[/latex]

8. [latex]2, -4, -\frac{3}{2}[/latex]

9. 4, –4, –5

10. [latex]\frac{1}{2}, \frac{1+\sqrt{5}}{2}, \frac{1-\sqrt{5}}{2}[/latex]

11. [latex]\frac{3}{2}[/latex]

12. 2, 3, –1, –2

13. [latex]-1, -1, \sqrt{5}, -\sqrt{5}[/latex]

14. [latex]2, 3+2i, 3 - 2i[/latex]

15. [latex]-\frac{1}{2}, 1+4i, 1 - 4i[/latex]

16. [latex]\pm 5, \pm 1, \pm \frac{5}{2}[/latex]

17. [latex]\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{6}[/latex]

18. [latex]1, \frac{1}{2}, -\frac{1}{3}[/latex]

19. [latex]2, \frac{1}{4}, -\frac{3}{2}[/latex]

20. [latex]\frac{5}{4}[/latex]

21. 8 by 4 by 6 inches

22. 8 by 5 by 3 inches

Inverses and Radical Functions Solutions

1. It can be too difficult or impossible to solve for x in terms of y.

2. We will need a restriction on the domain of the answer.

3. [latex]{f}^{-1}\left(x\right)=\sqrt{x}+4[/latex]

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

5. [latex]{f}^{-1}\left(x\right)=-\sqrt{\frac{x - 5}{3}}[/latex]

6. [latex]f\left(x\right)=\sqrt{9-x}[/latex]

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

8. [latex]{f}^{-1}\left(x\right)=\sqrt[3]{x - 5}[/latex]

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

10. [latex]{f}^{-1}\left(x\right)={\left(\frac{x - 9}{2}\right)}^{3}[/latex]

11. [latex]{f}^{-1}\left(x\right)={\frac{2 - 8x}{x}}[/latex]

12. [latex]{f}^{-1}\left(x\right)=\frac{7x - 3}{1-x}[/latex]

13. [latex]{f}^{-1}\left(x\right)=\sqrt{x}+4[/latex]
Graph of f(x)= (x-4)^2 and its inverse, f^(-1)(x)= sqrt(x)+4.

14. [latex]{f}^{-1}\left(x\right)=\sqrt[3]{1-x}[/latex]
Graph of f(x)= 1-x^3 and its inverse, f^(-1)(x)= (1-x)^(1/3).

15. [latex]{f}^{-1}\left(x\right)=\sqrt{\frac{1}{x}}[/latex]
Graph of f(x)= 1/x^2 and its inverse, f^(-1)(x)= sqrt(1/x).

16. [latex]t\left(h\right)=\sqrt{\frac{200-h}{4.9}}[/latex], 5.53 seconds

17. [latex]r\left(V\right)=\sqrt[3]{\frac{3V}{4\pi }}[/latex], 3.63 feet