# 9.4: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

## Exercise $$\PageIndex{1}$$

Which of the graphs below could be the graphs of a polynomial?

1. yes
2. no (due to the discontinuity)
3. no (due to horizontal asymptote)
4. no (due to corner)
5. yes (polynomial of degree 1)
6. yes

## Exercise $$\PageIndex{2}$$

Identify each of the graphs (a)-(e) with its corresponding assignment from (i)-(vi) below.

1. $$f(x)=-x^2$$
2. $$f(x)=-0.2x^2+1.8$$
3. $$f(x)=-0.6 x +3.8$$
4. $$f(x)=-0.2 x^3+0.4x^2+x-0.6$$
5. $$f(x)=x^3-6x^2+11x-4$$
6. $$f(x)=x^4$$
1. corresponds to (iii)
2. corresponds to (v)
3. corresponds to (vi)
4. corresponds to (ii)
5. corresponds to (iv)

## Exercise $$\PageIndex{3}$$

Identify the graph with its assignment below.

1. $$f(x)=x^6-14x^5+78.76 x^4-227.5 x^3+355.25x^2-283.5x+93$$
2. $$f(x)=-2x^5+30x^4-176x^3+504x^2-704x+386$$
3. $$f(x)=x^5-13x^4+65x^3-155x^2+174x-72$$
1. corresponds to (iii)
2. corresponds to (i)
3. corresponds to (ii)

## Exercise $$\PageIndex{4}$$

Sketch the graph of the function with the TI-84, which includes all extrema and intercepts of the graph.

1. $$f(x)=0.002 x^3+0.2 x^2-0.05x-5$$
2. $$f(x)=x^3+4x+50$$
3. $$f(x)=0.01 x^4-0.101 x^3-3 x^2+50.3x$$
4. $$f(x)=x^3-.007x$$
5. $$f(x)=x^3+.007x$$
6. $$f(x)=0.025 x^4+0.0975 x^3-1.215 x^2+2.89 x-22$$

## Exercise $$\PageIndex{4}$$

Find the exact value of at least one root of the given polynomial.

1. $$f(x)=x^3-10x^2+31x-30$$
2. $$f(x)=-x^3-x^2+8x+8$$
3. $$f(x)=x^3-11x^2-3x+33$$
4. $$f(x)=x^4+9x^3-6x^2-136x-192$$
5. $$f(x)=x^2+6x+3$$
6. $$f(x)=x^4-6x^3+3x^2+5x$$
1. $$x = 2$$, $$x = 3$$, or $$x = 5$$
2. $$x = −1$$
3. $$x = 11$$
4. $$x = −8$$, $$x = −3$$, $$x = −2$$, or $$x = 4$$
5. $$x=-3 \pm \sqrt{6}$$ (use the quadratic formula)
6. $$x = 0$$

## Exercise $$\PageIndex{5}$$

Graph the following polynomials without using the calculator.

1. $$f(x)=(x+4)^2(x-5)$$
2. $$f(x)=-3(x+2)^3 x^2 (x-4)^5$$
3. $$f(x)=2(x-3)^2(x-5)^3(x-7)$$
4. $$f(x)=-(x+4)(x+3)(x+2)^2(x+1)(x-2)^2$$