5.4E Exercises
- Page ID
- 155154
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Divide \(p(x)\) by \(d(x)\) and write the result in both forms: \( p(x) = q(x) \cdot d(x) + r(x) \) and \( \dfrac{p(x)}{d(x)} = q(x) + \dfrac{ r(x)}{d(x)} \).
- \( p(x) = 2x^3 + 5x^2 + x - 2 \), \(d(x) = x+2 \)
- \( p(x) = x^4 - x^3 - 9x + 9 \), \( d(x) = x-1\)
- \( p(x) = 4x^3 + 4x^2 + 11x + 8 \), \( d(x) = x^2 + 2 \)
- \( p(x) = x^5 + x^3 + x^2 + 5 \), \( d(x) = x^2 + 1 \)
- Answer
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- \( q(x) = 2x^2 + x - 1 \), \( r(x) = 0\).
- \( q(x) = x^3 - 9 \), \(r(x) = 0\).
- \( q(x) = 4x+4 = 4(x+1) \), \( r(x) = 3x \).
- \( q(x) = x^3 + 1 \), \(r(x) = 4 \).
A.) List all rational numbers that could be roots of \(f\). Test options until you find one that is indeed a root.
- \(f(x) = 5x^4 - 4x^3 + 3x^2 - 2x - 2 \)
- \( f(x) = 6x^3 + 23x^2 - 6x -8 \)
- \( f(x) = 3x^3 - 5x^2 - 11x + 18 \)
B.) Which of the following polynomial functions might have \( x = \dfrac{ 3}{4}\) as a root?
- \( f(x) = 7x^3 + 2x^2 - x + 3 \)
- \( g(x) = 8x^4 - 6x + 6 \)
- \( h(x) = 32x^5 + 2x^4 - x^2 - x + 15 \)
- \( p(x) = 4x^6 + 7x^3 + 5 \)
- Answer
-
A.)
- \( \pm 1, \pm 2, \pm \dfrac{1}{5}, \pm \dfrac{2}{5} \), \(x = 1 \).
- \( \pm 1, \pm 2, \pm 4, \pm 8, \pm \dfrac{1}{2}, \pm \dfrac{2}{3}, \pm \dfrac{4}{3}, \pm \dfrac{8}{3}, \pm \dfrac{1}{6} \), \(x = -\dfrac{1}{2} \).
- \( \pm 1, \pm 2, \pm 3, \pm 6, \pm 9, \pm 18, \pm \dfrac{1}{3}, \pm \dfrac{2}{3} \), \( x = 2 \).
B.) \(g(x)\) and \(h(x)\)
Factor the polynomial function (using any method you prefer, there may be shortcuts) and find all real roots.
- \( f(x) = 3x^3 - 6x^2 - 27x + 54 \)
- \( g(x) = 4x^4 - 1 \)
- \( h(x) = x^4 - 5x^2 + 4 \)
- \( p(x) = x^3+7x^2+16x + 12 \)
- \( q(x) = x^4 - 6x^3 + 10x^2 - 6x + 9 \)
- Answer
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- (Grouping or RRT + division) \( f(x) = 3(x-3)(x+3)(x-2) \), roots \( x = 2, \pm 3 \).
- (Difference of squares) \( g(x) = (2x^2 +1)(2x^2 - 1) \), first factor does not give real roots, second factor gives real roots \( x = \pm \sqrt{\frac{1}{2}} \).
- (Factoring trick or RRT + Division) \( h(x) = (x+1)(x-1)(x+2)(x-2)\), roots \( x = \pm 1, \pm 2 \).
- (RRT + Division) \( p(x) = (x+2)^2(x+3) \), roots \( x = -2\) (mult 2), \(x = -3\).
- (RRT + Division) \( q(x) = (x-3)^2(x^2 + 1) \), roots \(x = 3\) (mult 2), second factor does not give real roots.