This algebra 2 rational test answer key gives the full solution and the scoring guide for each of the 8 test questions on rational expressions and equations.
This answer key gives a worked solution and a scoring guide for each of the 8 questions. Award points as listed, and give partial credit when the method is correct but a sign or arithmetic slip appears.
Algebra 2 Rational Test Answer Key: questions 1–5
Question 1: Simplify and restrict (2 points)
\(\frac{(x-4)(x+3)}{(x-4)(x+4)}=\frac{x+3}{x+4}\). The original denominator is \((x-4)(x+4)\), so \(x\neq4\) and \(x\neq-4\).
\(\frac{x+3}{x+4}\), \(x\neq\pm4\)
Scoring: 1 point for the simplified expression, 1 point for both excluded values.
Question 2: Divide rational expressions (2 points)
Multiply by the reciprocal: \(\frac{(x-5)(x+5)}{x(x+2)}\cdot\frac{x+2}{x+5}=\frac{x-5}{x}\). Excluded values: \(x\neq0,\ -2,\ -5\).
\(\frac{x-5}{x}\), \(x\neq0,\ -2,\ -5\)
Scoring: 1 point for factoring and using the reciprocal, 1 point for the simplified result with the excluded values.
Question 3: Add and subtract (3 points)
a) LCD \((x+2)(x-1)\): \(\frac{3(x-1)+4(x+2)}{(x+2)(x-1)}=\frac{3x-3+4x+8}{(x+2)(x-1)}=\frac{7x+5}{(x+2)(x-1)}\).
b) LCD \(x^{2}-4=(x-2)(x+2)\): \(\frac{x-2(x+2)}{(x-2)(x+2)}=\frac{x-2x-4}{(x-2)(x+2)}=\frac{-x-4}{(x-2)(x+2)}\).
a) \(\frac{7x+5}{(x+2)(x-1)}\) b) \(\frac{-(x+4)}{(x-2)(x+2)}\)
Scoring: part a, 1 point. Part b, 1 point for the correct LCD and rewriting, 1 point for the correct numerator.
Question 4: Rewrite a rational expression (2 points)
Divide: \(x^{2}: x=x\); subtract \(x^{2}+2x\) to get \(3x+1\); \(3x: x=3\); subtract \(3x+6\) to get \(-5\).
\(x+3-\frac{5}{x+2}\)
Scoring: 1 point for the quotient \(x+3\), 1 point for the remainder term \(-\frac{5}{x+2}\).
Question 5: Solve and check (3 points)
a) Multiply by \(x-4\): \(x^{2}=3x+4\), so \(x^{2}-3x-4=0\) and \((x-4)(x+1)=0\). Then \(x=4\) or \(x=-1\).
b) At \(x=4\) the denominators are \(0\), so \(x=4\) is extraneous. Check \(x=-1\): \(\frac{1}{-5}=\frac{-3+4}{-5}=\frac{1}{-5}\).
a) \(x=4\) or \(x=-1\) b) \(x=4\) is extraneous; the solution is \(x=-1\)
Scoring: part a, 1 point for \(x^{2}-3x-4=0\) and 1 point for both values. Part b, 1 point.
Algebra 2 Rational Test Answer Key: questions 6–8
Question 6: Equation and graph (2 points)
a) Multiply by \(x\): \(12=x^{2}+x\), so \(x^{2}+x-12=0\) and \((x+4)(x-3)=0\). Then \(x=-4\) or \(x=3\). Both are allowed because neither is \(0\).
b) At \(x=3\), \(y=4\). At \(x=-4\), \(y=-3\).
a) \(x=-4\) or \(x=3\) b) \((3,4)\) and \((-4,-3)\)
Scoring: 1 point for each part.
Question 7: Washing the vans (3 points)
a) Diego does \(\frac{1}{10}\) of the job per hour and Liam does \(\frac{1}{15}\). Together \(\frac{1}{10}+\frac{1}{15}=\frac{1}{t}\).
b) Multiply by \(30t\): \(3t+2t=30\), so \(5t=30\) and \(t=6\). Check: \(\frac{6}{10}+\frac{6}{15}=\frac{3}{5}+\frac{2}{5}=1\).
a) \(\frac{1}{10}+\frac{1}{15}=\frac{1}{t}\) b) \(6\) hours
Scoring: part a, 1 point. Part b, 1 point for clearing denominators correctly and 1 point for the answer.
Question 8: Road trip (3 points)
a) \(\frac{120}{r}+\frac{60}{r-20}=3.5\).
b) Multiply by \(2r(r-20)\): \(240(r-20)+120r=7r(r-20)\), so \(360r-4800=7r^{2}-140r\). Then \(7r^{2}-500r+4800=0\), which factors as \((r-60)(7r-80)=0\).
So \(r=60\) or \(r=\frac{80}{7}\approx11.4\). The town speed \(r-20\) must be positive, so \(\frac{80}{7}\) is rejected. Check \(r=60\): \(\frac{120}{60}+\frac{60}{40}=2+1.5=3.5\).
a) \(\frac{120}{r}+\frac{60}{r-20}=3.5\) b) \(r=60\) mph
Scoring: part a, 1 point. Part b, 1 point for the quadratic and 1 point for \(r=60\) with the other root rejected.
Scoring summary
20 points: add the points of every question of this algebra 2 rational test answer key.
How to use this algebra 2 rational test answer key
Use this algebra 2 rational test answer key to score the test fairly and to see which skills need more practice.
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Algebra 2 Rational Expressions and Equations test answer key
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