8.12: Exercise Supplement
- Page ID
- 60054
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Exercise Supplement
Rational Expressions
For the following problems, find the domain of each rational expression.
\(\dfrac{9}{x+4}\)
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\(x \not = 4\)
\(\dfrac{10x}{x+6}\)
\(\dfrac{x+1}{2x - 5}\)
- Answer
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\(x \not = \dfrac{5}{2}\)
\(\dfrac{2a + 3}{7a + 5}\)
\(\dfrac{3m}{2m(m-1)}\)
- Answer
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\(m \not = 0, 1\)
\(\dfrac{5r + 6}{9r(2r + 1)}\)
\(\dfrac{s}{s(s+8)(4s + 7)}\)
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\(s \not = -8, -\dfrac{7}{4}, 0\)
\(\dfrac{-11x}{x^2 - 9x + 18}\)
\(\dfrac{-y + 5}{12y^2 + 28y - 5}\)
- Answer
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\(y \not = \dfrac{1}{6}, -\dfrac{5}{2}\)
\(\dfrac{16}{12a^3 + 21a^2 - 6a}\)
For the following problems, show that the fractions are equivalent.
\(\dfrac{-4}{5}, -\dfrac{4}{5}\)
- Answer
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\(-(4 \cdot 5) = 20, -4(5) = -20\)
\(\dfrac{-3}{8}, -\dfrac{3}{8}\)
\(\dfrac{-7}{10}, -\dfrac{7}{10}\)
- Answer
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\(−(7 \cdot 10)=−70,−7(10)=−70\)
For the following problems, fill in the missing term.
\(-\dfrac{3}{y-5} = \dfrac{?}{y - 5}\)
\(-\dfrac{6a}{2a + 1} = \dfrac{?}{2a + 1}\)
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\(−6a\)
\(-\dfrac{x+1}{x-3} = \dfrac{?}{x-3}\)
\(-\dfrac{9}{-a + 4} = \dfrac{?}{a - 4}\)
- Answer
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\(9\)
\(\dfrac{y + 3}{-y-5} = \dfrac{?}{y + 5}\)
\(\dfrac{-6m-7}{-5m-1} = \dfrac{6m + 7}{?}\)
- Answer
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\(5m+1\)
\(-\dfrac{2r - 5}{7r + 1} = \dfrac{2r - 5}{?}\)
Reducing Rational Expressions
For the following problems, reduce the rational expressions to lowest terms.
\(\dfrac{12}{6x + 24}\)
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\(\dfrac{2}{x + 4}\)
\(\dfrac{16}{4y - 16}\)
\(\dfrac{5m + 25}{10m^2 + 15m}\)
- Answer
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\(\dfrac{m+5}{m(2m + 3)}\)
\(\dfrac{7 + 21r}{7r^2 + 28r}\)
\(\dfrac{3a^2 + 4a}{5a^3 + 6a^2}\)
- Answer
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\(\dfrac{3a + 4}{a(5a + 6)}\)
\(\dfrac{4x - 4}{x^2 + 2x - 3}\)
\(\dfrac{5y + 20}{y^2 - 16}\)
- Answer
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\(\dfrac{5}{y-4}\)
\(\dfrac{4y^3 - 12}{y^4 - 2y^2 - 3}\)
\(\dfrac{6a^9 - 12a^7}{2a^7 - 14a^5}\)
- Answer
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\(\dfrac{3a^2(a^2 - 2)}{a^2 - 7}\)
\(\dfrac{8x^4y^8 + 24x^3y^9}{4x^2y^5 - 12x^3y^6}\)
\(\dfrac{21y^8z^{10}w^2}{-7y^7w^2}\)
- Answer
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\(-3yz^{10}\)
\(\dfrac{-35a^5b^2c^4d^8}{-5abc^3d^6}\)
\(\dfrac{x^2 + 9x + 18}{x^3 + 3x^2}\)
- Answer
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\(\dfrac{x+6}{x^2}\)
\(\dfrac{a^2 - 12a + 35}{2a^4 - 14a^3}\)
\(\dfrac{y^2 - 7y + 12}{y^2 - 4y + 3}\)
- Answer
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\(\dfrac{y-4}{y-1}\)
\(\dfrac{m^2 - 6m - 16}{m^2 - 9m - 22}\)
\(\dfrac{12r^2 - 7r - 10}{4r^2 - 13r + 10}\)
- Answer
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\(\dfrac{3r + 2}{r - 2}\)
\(\dfrac{14a^2 - 5a - 1}{6a^2 + 9a - 6}\)
\(\dfrac{4a^4 - 8a^3}{4a^2}\)
- Answer
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\(a(a-2)\)
\(\dfrac{5m^2}{10m^3 + 5m^2}\)
\(\dfrac{-6a - 1}{-5a - 2}\)
- Answer
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\(\dfrac{6a + 1}{5a + 2}\)
\(\dfrac{-r}{-5r - 1}\)
Multiplying and Dividing Rational Expressions - Adding and Subtracting Rational Expressions
For the following problems, perform the indicated operations.
\(\dfrac{x^2}{18} \cdot \dfrac{3}{x^3}\)
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\(\dfrac{1}{6x}\)
\(\dfrac{4a^2b^3}{15x^4y^5} \cdot \dfrac{10x^6y^3}{ab^2}\)
\(\dfrac{x+6}{x-1} \cdot \dfrac{x+7}{x+6}\)
- Answer
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\(\dfrac{x+7}{x-1}\)
\(\dfrac{8a - 12}{3a + 3} \div \dfrac{(a+1)^2}{4a - 6}\)
\(\dfrac{10m^4 - 5m^2}{4r^7 + 20r^3} \div \dfrac{m}{16r^8 + 80r^4}\)
- Answer
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\(20mr(2m^2 - 1)\)
\(\dfrac{5}{r + 7} - \dfrac{3}{r + 7}\)
\(\dfrac{2a}{3a - 1} - \dfrac{9a}{3a - 1}\)
- Answer
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\(\dfrac{-7a}{3a - 1}\)
\(\dfrac{9x + 7}{4x - 6} + \dfrac{3x + 2}{4x - 6}\)
\(\dfrac{15y - 4}{8y + 1} - \dfrac{2y + 1}{8y + 1}\)
- Answer
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\(\dfrac{13y - 5}{8y + 1}\)
\(\dfrac{4}{a + 3} + \dfrac{6}{a - 5}\)
\(\dfrac{7a}{a + 6} + \dfrac{5a}{a-8}\)
- Answer
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\(\dfrac{2a(6a - 13)}{(a+6)(a-8)}\)
\(\dfrac{x+4}{x-2} - \dfrac{y+6}{y+1}\)
\(\dfrac{2y + 1}{y + 4} - \dfrac{y + 6}{y + 1}\)
- Answer
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\(\dfrac{y^2 - 7y - 23}{(y+4)(y+1)}\)
\(\dfrac{x-3}{(x+2)(x+4)} + \dfrac{2x - 1}{x + 4}\)
\(\dfrac{6a + 5}{(2a + 1)(4a - 3)} + \dfrac{4a + 1}{2a + 1}\)
- Answer
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\(\dfrac{2(8a^2 - a + 1)}{(2a + 1)(4a - 3)}\)
\(\dfrac{4}{x^2 + 3x + 2} + \dfrac{9}{x^2 + 6x + 8}\)
\(\dfrac{6r}{r^2 + 7r - 18} - \dfrac{-3r}{r^2 - 3r + 2}\)
- Answer
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\(\dfrac{3r(3r + 7)}{(r-1)(r-2)(r+9)}\)
\(\dfrac{y+3}{y^2 - 11y + 10} - \dfrac{y + 1}{y^2 + 3y - 4}\)
\(\dfrac{2a + 5}{16a^2 - 1} - \dfrac{6a + 7}{16a^2 - 12a + 2}\)
- Answer
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\(\dfrac{-16a^2 - 18a - 17}{2(4a-1)(4a+1)(2a-1)}\)
\(\dfrac{7y + 4}{6y^2 - 32y + 32} + \dfrac{6y - 10}{2y^2 - 18y + 40}\)
\(\dfrac{x^2 - x - 12}{x^2 - 3x + 2} \cdot \dfrac{x^2 + 3x - 4}{x^2 - 3x - 18}\)
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\(\dfrac{(x+4)(x-4)}{(x-2)(x-6)}\)
\((r+3)^4 \cdot \dfrac{r+4}{(r+3)^3}\)
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\((r+3)(r+4)\)
\((b+5)^3 \cdot \dfrac{(b+1)^2}{(b+5)^2}\)
\((x-7)^4 \div \dfrac{(x-7)^3}{x+1}\)
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\((x-7)(x+1)\)
\((4x + 9)^6 \div \dfrac{(4x + 9)^2}{(3x + 1)^4}\)
\(5x + \dfrac{2x^2 + 1}{x-4}\)
- Answer
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\(\dfrac{7x^2 - 20x + 1}{(x-4)}\)
\(2y + \dfrac{4y^2 + 5}{y - 1}\)
\(\dfrac{y^2 + 4y + 4}{y^2 + 10y + 21} \div (y+2)\)
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\(\dfrac{(y+2)}{(y+3)(y+7)}\)
\(2x - 3 + \dfrac{4x^2 + x - 1}{x - 1}\)
\(\dfrac{3 x+1}{x^{2}+3 x+2}+\dfrac{5 x+6}{x^{2}+6 x+5}-\dfrac{3 x-7}{x^{2}-2 x-35}\)
- Answer
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\(\dfrac{5 x^{3}-26 x^{2}-192 x-105}{\left(x^{2}-2 x-35\right)(x+1)(x+2)}\)
\(\dfrac{5 a+3 b}{8 a^{2}+2 a b-b^{2}}-\dfrac{3 a-b}{4 a^{2}-9 a b+2 b^{2}}-\dfrac{a+5 b}{4 a^{2}+3 a b-b^{2}}\)
\(\dfrac{3 x^{2}+6 x+10}{10 x^{2}+11 x-6}+\dfrac{2 x^{2}-4 x+15}{2 x^{2}-11 x-21}\)
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\(\dfrac{13 x^{3}-39 x^{2}+51 x-100}{(2 x+3)(x-7)(5 x-2)}\)
\(\dfrac{y^2 - 1}{y^2 + 9y + 20} \div \dfrac{y^2 + 5y - 6}{y^2 - 16}\)
Rational Equations
For the following problems, solve the rational equations.
\(\dfrac{4x}{5} + \dfrac{3x - 1}{15} = \dfrac{29}{25}\)
\(\dfrac{6a}{7} + \dfrac{2a - 3}{21} = \dfrac{77}{21}\)
- Answer
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\(a = 4\)
\(\dfrac{5x - 1}{6} + \dfrac{3x + 4}{9} = \dfrac{-8}{9}\)
\(\dfrac{4y - 5}{4} + \dfrac{8y + 1}{6} = \dfrac{-69}{12}\)
- Answer
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\(y=−2\)
\(\dfrac{4}{x-1} + \dfrac{7}{x+2} = \dfrac{43}{x^2 + x - 2}\)
\(\dfrac{5}{a + 3} + \dfrac{6}{a - 4} = \dfrac{9}{a^2 - a - 12}\)
- Answer
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\(a = 1\)
\(\dfrac{-5}{y - 3} + \dfrac{2}{y - 3} = \dfrac{3}{y - 3}\)
\(\dfrac{2m + 5}{m - 8} + \dfrac{9}{m - 8} = \dfrac{30}{m - 8}\)
- Answer
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No solution; \(m=8\) is excluded.
\(\dfrac{r + 6}{r - 1} - \dfrac{3r + 2}{r - 1} = \dfrac{-6}{r - 1}\)
\(\dfrac{8b + 1}{b-7} - \dfrac{b + 5}{b - 7} = \dfrac{45}{b - 7}\)
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No solution; \(b=7\) is excluded.
Solve \(z = \dfrac{x - \hat{x}}{x}\) for \(s\)
Solve \(A = P(1 + rt)\) for \(t\).
- Answer
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\(t = \dfrac{A-P}{Pr}\)
Solve \(\dfrac{1}{R} = \dfrac{1}{E} + \dfrac{1}{F}\) for \(E\).
Solve \(Q = \dfrac{2mn}{s + t}\) for \(t\)
- Answer
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\(t = \dfrac{2mn - Qs}{Q}\)
Solve \(I = \dfrac{E}{R + r}\) for \(r\)
Applications
For the following problems, find the solution.
When the same number is subtracted from both terms of the fraction \(\dfrac{7}{12}\), the result is \(\dfrac{1}{2}\). What is the number?
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\(2\)
When the same number is added to both terms of the fraction \(\dfrac{13}{15}\), the result is \(\dfrac{8}{9}\). What is the number?
When three-fourths of a number is added to the reciprocal of the number, the result is \(\dfrac{173}{16}\). What is the number?
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No rational solution.
When one-third of a number is added to the reciprocal of the number, the result is \(\dfrac{127}{90}\). What is the number?
Person A working alone can complete a job in 9 hours. Person B working alone can complete the same job in 7 hours. How long will it take both people to complete the job working together?
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\(3\dfrac{15}{16}\) hrs.
Debbie can complete an algebra assignment in \(\dfrac{3}{4}\) of an hour. Sandi, who plays her radio while working, can complete the same assignment in \(1\dfrac{1}{4}\) hours. If Debbie and Sandi work together, how long will it take them to complete the assignment?
An inlet pipe can fill a tank in 6 hours and an outlet pipe can drain the tank in 8 hours. If both pipes are open, how long will it take to fill the tank?
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24 hrs
Two pipes can fill a tank in 4 and 5 hours, respectively. How long will it take both pipes to fill the tank?
The pressure due to surface tension in a spherical bubble is given by \(P = \dfrac{4T}{r}\), where \(T\) is the surface tension of the liquid, and \(r\) is the radius of the bubble.
(a) Determine the pressure due to surface tension within a soap bubble of radius \(\dfrac{1}{2}\) inch and surface tension 22.
(b) Determine the radius of a bubble if the pressure due to surface tension is 57.6 and the surface tension is 18.
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a) 176 units of pressure
b) \(\dfrac{5}{4}\) units of length.
The equation \(\dfrac{1}{p} + \dfrac{1}{q} = \dfrac{1}{f} \) relates an object's distance \(p\) from a lens and the image distance \(q\) from the lens to the focal length \(f\) of the lens.
(a) Determine the focal length of a lens in which an object 8 feet away produces an image 6 feet away.
(b) Determine how far an object is from a lens if the focal length of the lens is 10 inches and the image distance is 10 inches.
(c) Determine how far an object will be from a lens that has a focal length of \( \dfrac{7}{8}\) cm and the object distance is 3 cm away from the lens.
Dividing Polynomials
For the following problems, divide the polynomials.
\(a^2 + 9a + 18\) by \(a + 3\)
- Answer
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\(a+6\)
\(c^2 + 3c - 88\) by \(c - 8\)
\(x^3 + 9x^2 + 18x + 28\) by \(x + 7\).
- Answer
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\(x^2 + 2x + 4\)
\(9y^3 - 2y^2 - 49y - 6\) by \(y + 6\)
\(m^4 + 2m^3 - 8m^2 - m + 2\) by \(m - 2\).
- Answer
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\(m^3 + 4m^2 - 1\)
\(3r^2 - 17r - 27\) by \(r - 7\)
\(a^3 - 3a^2 - 56a + 10\) by \(a-9\).
- Answer
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\(a^2 + 6a - 2 - \dfrac{8}{a-9}\)
\(x^3 - x + 1\) by \(x + 3\)
\(y^3 + y^2 - y\) by \(y + 4\)
- Answer
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\(y^2 - 3y + 11 - \dfrac{44}{y + 4}\)
\(5x^6 + 5x^5 - 2x^4 + 5x^3 - 7x^2 - 8x + 6\) by \(x^2 + x - 1\)
\(y^{10} - y^7 + 3y^4 - 3y\) by \(y^4 - y\)
- Answer
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\(y^6 + 3\)
\(-4b^7 - 3b^6 - 22b^5 - 19b^4 + 12b^3 - 6b^2 + b + 4\) by \(b^2 + 6\).
\(x^3 + 1\) by \(x + 1\)
- Answer
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\(x^2 - x + 1\)
\(a^4 + 6a^3 + 4a^2 + 12a + 8\) by \(a^2 + 3a + 2\)
\(y^{10} + 6y^5 + 9\) by \(y^5 + 3\)
- Answer
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\(y^5 + 3\)