4.1E: An Introduction to The Graphs of Polynomial Functions (Exercises)
- Page ID
- 99731
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)In # 1- 5 find all the zeros of each polynomial (and state their multiplicity)
1. \(f(x)=2(x-2)(x+3)^3\)
2. \(f(x)=4x^6+9x^4\)
3. \(f(x)=8x^3+6x^2-27x\)
4. \(f(x)=2x^4+5x^3-3x^2\)
5. \(f(x)=x^3+x^2+x\)
6. Find a formula for a polynomial with the zeros \(x=-3\), \(x=0\), \(x=1\), and \(x=2\) with multiplicity 1 and whose graph contains the point (3,4).
7. Find a formula for a polynomial with the zero \(x=-2\) with multiplicity 2, the zero \(x=0\) with multiplicity 3, and whose graph contains the point (1,-3).
8. Find a formula for a polynomial with the zero \(x=4\) with multiplicity 3, the zero \(x=-1\) with multiplicity 1, and whose graph contains the point (0,5).
9. Find a formula for a polynomial with real coefficients that has the zero \(x=3i\) with multiplicity 1, the zero \(x=2\) with multiplicity 2, and whose graph contains the point (1,5).
10. Find a formula for a polynomial with real coefficients that has the zero \(x=2-i\) with multiplicity 1, the zero \(x=0\) with multiplicity 3, and whose graph contains the point (-2,4).
11. What is the degree of the polynomial \(f(x)=6+2x +1x^5+x^4\)? How many possible distinct zeros can /(f/) have?
12. Determine if the statement is true or false. Justify your conclusion. "A polynomial of degree 7 may have 8 distinct zeros".
13. Determine if the statement is true or false. Justify your conclusion. "A polynomial of degree 7 may have 6 distinct zeros".
- Answer
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2. The zeros are \(x=0\) with multiplicity 4, \(x=\dfrac{3}{2}\) with multiplicity 1, and \(x=-\dfrac{3}{2}\) with multiplicity 1
7. \(y=-\dfrac{1}{3}x^3(x+2)^2 \)
9. \(y=\dfrac{1}{2}(x^2+9)(x-2)^2 \)