3.3.1: Exercises 3.3
- Page ID
- 63465
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Terms and Concepts
Exercise \(\PageIndex{1}\)
Can the fraction \(\displaystyle \frac{x+2}{x^2+2}\) be simplified? Explain.
- Answer
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No, the numerator and denominator have no common factors.
Exercise \(\PageIndex{2}\)
In the fraction \(\displaystyle \frac{2}{(x+3)^2(x+2)}\) are there any repeated factors? If so, what factor(s) are repeated, and how many times?
- Answer
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Yes, \(x+3\) is repeated twice
Exercise \(\PageIndex{3}\)
What is meant by an irreducible quadratic?
- Answer
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A quadratic that has no real valued roots
Exercise \(\PageIndex{4}\)
Give an example of an irreducible quadratic.
- Answer
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Answers will vary; \(x^2+a\) is an example if \(a>0\)
Problems
Simplify the given expression in ecercises \(\PageIndex{5}\) - \(\PageIndex{9}\).
Exercise \(\PageIndex{5}\)
\(\displaystyle \frac{5}{18} - \frac{5}{12}\)
- Answer
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\(\displaystyle \frac{-5}{36}\)
Exercise \(\PageIndex{6}\)
\(\displaystyle \frac{x}{b} - \frac{b}{x}\)
- Answer
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\(\displaystyle \frac{x^2-b^2}{xb}\)
Exercise \(\PageIndex{7}\)
\(\displaystyle \frac{x}{y^2} - \frac{x}{x+y}\)
- Answer
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\(\displaystyle \frac{x^2+xy-xy^2}{xy^2 + y^3}\)
Exercise \(\PageIndex{8}\)
\(\displaystyle \frac{\phantom{x} \frac{1}{x} - \frac{x+2}{x^2} \phantom{x}}{\frac{4}{x^2} - \frac{x^2+1}{x^3}}\)
- Answer
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\(\displaystyle \frac{\phantom{x} -2x}{-x^2+4x-1 \phantom{x} }\), \(x \neq 0\)
Exercise \(\PageIndex{9}\)
\(\displaystyle \frac{\phantom{x} \frac{1}{x-b} - \frac{1}{x} \phantom{x}}{b}\)
- Answer
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\(\displaystyle \frac{1}{x^2-bx}\), \(b\neq 0\)
In exercises \(\PageIndex{10}\) - \(\PageIndex{16}\), decompose the given fraction. Do not solve for \(A\), \(B\), etc.
Exercise \(\PageIndex{10}\)
\(\displaystyle \frac{x-8}{(x+2)^3}\)
- Answer
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\(\displaystyle \frac{A}{x+2} + \frac{B}{(x+2)^2} + \frac{C}{(x+2)^3}\)
Exercise \(\PageIndex{11}\)
\(\displaystyle \frac{4}{(s-1)^2(2s-5)(s+3)}\)
- Answer
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\(\displaystyle \frac{A}{s-1} + \frac{B}{(s-1)^2} + \frac{C}{2s-5} + \frac{D}{s+3}\)
Exercise \(\PageIndex{12}\)
\(\displaystyle \frac{5t^2+11t-9}{(t+1)^3(t^2+1)^2}\)
- Answer
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\(\displaystyle \frac{A}{t+1} + \frac{B}{(t+1)^2} + \frac{C}{(t+1)^3} + \frac{Dt+E}{t^2+1} + \frac{Ft+G}{(t^2+1)^2}\)
Exercise \(\PageIndex{13}\)
\(\displaystyle \frac{6x}{(x-4)(x^2+x+5)}\)
- Answer
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\(\displaystyle \frac{A}{x-4} + \frac{Bx+C}{x^2+x+5}\)
Exercise \(\PageIndex{14}\)
\(\displaystyle \frac{3x-7}{x^4-1}\)
- Answer
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\(\displaystyle \frac{A}{x+1} + \frac{B}{x-1} + \frac{Cx+D}{x^2+1}\)
Exercise \(\PageIndex{15}\)
\(\displaystyle \frac{2s}{s^3+1}\)
- Answer
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\(\displaystyle \frac{A}{s+1} + \frac{Bs+C}{s^2-s+1}\)
Exercise \(\PageIndex{16}\)
\(\displaystyle \frac{11}{t^2-6t+5}\)
- Answer
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\(\displaystyle \frac{A}{t-5} + \frac{B}{t-1}\)
In exercises \(\PageIndex{17}\) - \(\PageIndex{22}\), fully decompose the given fraction.
Exercise \(\PageIndex{17}\)
\(\displaystyle \frac{x+5}{x^2+x-2}\)
- Answer
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\(\displaystyle \frac{-1}{x+2} + \frac{2}{x-1}\)
Exercise \(\PageIndex{18}\)
\(\displaystyle \frac{1}{x^2-a^2}\)
- Answer
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\(\displaystyle \frac{1/(2a)}{x-a} - \frac{1/(2a)}{x+a}\)
Exercise \(\PageIndex{19}\)
\(\displaystyle \frac{2s^2-s+4}{s^3+4s}\)
- Answer
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\(\displaystyle \frac{1}{s} + \frac{s-1}{s^2+4}\)
Exercise \(\PageIndex{20}\)
\(\displaystyle \frac{y-1}{y^2+3y+2}\)
- Answer
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\(\displaystyle \frac{3}{y+2} - \frac{2}{y+1}\)
Exercise \(\PageIndex{21}\)
\(\displaystyle \frac{4x}{x^3-x^2-x+1}\)
- Answer
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\(\displaystyle \frac{-1}{x+1} + \frac{1}{x-1} + \frac{2}{(x-1)^2}\)
Exercise \(\PageIndex{22}\)
\(\displaystyle \frac{x^2+2x-1}{2x^3+3x^2-2x}\)
- Answer
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\(\displaystyle \frac{1/2}{x} + \frac{1/5}{2x-1} - \frac{1/10}{x+2}\)