ACCUPLACER Advanced Algebra & Functions Quiz: Simplifying Rational Expressions
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Simplifying Rational ExpressionsQuestion 1 of 20

Simplify the expression 5x−x2x2−25\frac{5x - x^2}{x^2 - 25}.

xx−5\frac{x}{x-5}
xx+5\frac{x}{x+5}
−xx−5\frac{-x}{x-5}
−xx+5\frac{-x}{x+5}
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ACCUPLACER Advanced Algebra & Functions Quiz

ACCUPLACER Advanced Algebra & Functions Quiz: Simplifying Rational Expressions

Practice Simplifying Rational Expressions in ACCUPLACER Advanced Algebra & Functions with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Simplifying Rational Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for ACCUPLACER Advanced Algebra & Functions.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Simplify the expression 5x−x2x2−25\frac{5x - x^2}{x^2 - 25}.

  1. xx−5\frac{x}{x-5}
  2. xx+5\frac{x}{x+5}
  3. −xx−5\frac{-x}{x-5}
  4. −xx+5\frac{-x}{x+5} (correct answer)
Explanation: First, factor the numerator and the denominator. Factor out xx from the numerator to get x(5−x)x(5-x). Factor the denominator, a difference of squares, to get (x−5)(x+5)(x-5)(x+5). The expression is x(5−x)(x−5)(x+5)\frac{x(5-x)}{(x-5)(x+5)}. The terms (5−x)(5-x) and (x−5)(x-5) are opposites. We can write (5−x)(5-x) as −1(x−5)-1(x-5). The expression becomes −x(x−5)(x−5)(x+5)\frac{-x(x-5)}{(x-5)(x+5)}. Cancel the common factor (x−5)(x-5) to get −xx+5\frac{-x}{x+5}.

Question 2

Simplify the expression 2x2−5x−3x2−5x+6\frac{2x^2 - 5x - 3}{x^2 - 5x + 6}.

  1. 2x−1x−3\frac{2x-1}{x-3}
  2. 2x+1x−2\frac{2x+1}{x-2} (correct answer)
  3. 2x+1x+2\frac{2x+1}{x+2}
  4. 2x−3x−3\frac{2x-3}{x-3}
Explanation: To simplify, factor both the numerator and the denominator. The numerator 2x2−5x−32x^2 - 5x - 3 factors into (2x+1)(x−3)(2x+1)(x-3). The denominator x2−5x+6x^2 - 5x + 6 factors into (x−2)(x−3)(x-2)(x-3). The expression becomes (2x+1)(x−3)(x−2)(x−3)\frac{(2x+1)(x-3)}{(x-2)(x-3)}. Canceling the common factor (x−3)(x-3) yields 2x+1x−2\frac{2x+1}{x-2}.

Question 3

Which expression is equivalent to x2−4y2x2−2xy\frac{x^2 - 4y^2}{x^2 - 2xy} for all values where it is defined?

  1. x−2yx\frac{x-2y}{x}
  2. x+2yx\frac{x+2y}{x} (correct answer)
  3. 1+2y1+2y
  4. x+2yx−2y\frac{x+2y}{x-2y}
Explanation: Factor the numerator and the denominator. The numerator x2−4y2x^2 - 4y^2 is a difference of squares, factoring to (x−2y)(x+2y)(x-2y)(x+2y). The denominator x2−2xyx^2 - 2xy has a greatest common factor of xx, factoring to x(x−2y)x(x-2y). The expression is (x−2y)(x+2y)x(x−2y)\frac{(x-2y)(x+2y)}{x(x-2y)}. Canceling the common factor (x−2y)(x-2y) leaves x+2yx\frac{x+2y}{x}.

Question 4

Which of the following is equivalent to 12−3x2x2−32\frac{12 - 3x}{2x^2 - 32}?

  1. 32(x+4)\frac{3}{2(x+4)}
  2. 32(x−4)\frac{3}{2(x-4)}
  3. −32(x+4)\frac{-3}{2(x+4)} (correct answer)
  4. −32(x−4)\frac{-3}{2(x-4)}
Explanation: First, factor the GCF from the numerator: 3(4−x)3(4-x). Then, factor the GCF from the denominator: 2(x2−16)2(x^2-16). The denominator can be factored further as a difference of squares: 2(x−4)(x+4)2(x-4)(x+4). The expression is 3(4−x)2(x−4)(x+4)\frac{3(4-x)}{2(x-4)(x+4)}. Recognize that (4−x)=−1(x−4)(4-x) = -1(x-4). Substitute this into the numerator: −3(x−4)2(x−4)(x+4)\frac{-3(x-4)}{2(x-4)(x+4)}. Cancel the common factor (x−4)(x-4) to get −32(x+4)\frac{-3}{2(x+4)}.

Question 5

Simplify the rational expression 2x3−2xx3+4x2−5x\frac{2x^3 - 2x}{x^3 + 4x^2 - 5x}.

  1. 2(x−1)x+5\frac{2(x-1)}{x+5}
  2. 2(x+1)x+5\frac{2(x+1)}{x+5} (correct answer)
  3. 2(x+1)x−5\frac{2(x+1)}{x-5}
  4. x+1x+5\frac{x+1}{x+5}
Explanation: First, factor out the greatest common factor from both the numerator and denominator. Numerator GCF is 2x2x, giving 2x(x2−1)2x(x^2-1). Denominator GCF is xx, giving x(x2+4x−5)x(x^2+4x-5). Then, factor the remaining polynomials. x2−1=(x−1)(x+1)x^2-1 = (x-1)(x+1) and x2+4x−5=(x+5)(x−1)x^2+4x-5 = (x+5)(x-1). The expression becomes 2x(x−1)(x+1)x(x+5)(x−1)\frac{2x(x-1)(x+1)}{x(x+5)(x-1)}. Cancel the common factors of xx and (x−1)(x-1). This leaves 2(x+1)x+5\frac{2(x+1)}{x+5}.

Question 6

Simplify the expression a2+ab−6b2a2−4b2\frac{a^2 + ab - 6b^2}{a^2 - 4b^2}.

  1. a+3ba+2b\frac{a+3b}{a+2b} (correct answer)
  2. a−3ba−2b\frac{a-3b}{a-2b}
  3. a+3ba−2b\frac{a+3b}{a-2b}
  4. ab−6b2a−4b2\frac{ab-6b^2}{a-4b^2}
Explanation: Factor the trinomial in the numerator: a2+ab−6b2=(a+3b)(a−2b)a^2 + ab - 6b^2 = (a+3b)(a-2b). Factor the denominator, which is a difference of squares: a2−4b2=(a−2b)(a+2b)a^2 - 4b^2 = (a-2b)(a+2b). The expression is (a+3b)(a−2b)(a−2b)(a+2b)\frac{(a+3b)(a-2b)}{(a-2b)(a+2b)}. Cancel the common factor (a−2b)(a-2b). The result is a+3ba+2b\frac{a+3b}{a+2b}.

Question 7

Simplify the expression 12x2+10x−88x2−2\frac{12x^2 + 10x - 8}{8x^2 - 2}.

  1. 3x+42x+1\frac{3x+4}{2x+1} (correct answer)
  2. 3x+42x−1\frac{3x+4}{2x-1}
  3. 6x−44x+1\frac{6x-4}{4x+1}
  4. 3x−22x−1\frac{3x-2}{2x-1}
Explanation: First, factor out the GCF from the numerator (2) and the denominator (2). This gives 2(6x2+5x−4)2(4x2−1)\frac{2(6x^2+5x-4)}{2(4x^2-1)}. The 2s cancel. Now factor the remaining polynomials. The numerator 6x2+5x−46x^2+5x-4 factors to (3x+4)(2x−1)(3x+4)(2x-1). The denominator 4x2−14x^2-1 is a difference of squares, factoring to (2x−1)(2x+1)(2x-1)(2x+1). The expression is (3x+4)(2x−1)(2x−1)(2x+1)\frac{(3x+4)(2x-1)}{(2x-1)(2x+1)}. Cancel the common factor (2x−1)(2x-1) to get 3x+42x+1\frac{3x+4}{2x+1}.

Question 8

Simplify the expression ab+3a−2b−6b2+6b+9\frac{ab+3a-2b-6}{b^2+6b+9}.

  1. a+2b+3\frac{a+2}{b+3}
  2. a−2b−3\frac{a-2}{b-3}
  3. a−2b+3\frac{a-2}{b+3} (correct answer)
  4. a−b−3b+3\frac{a-b-3}{b+3}
Explanation: Factor the numerator by grouping: a(b+3)−2(b+3)=(a−2)(b+3)a(b+3)-2(b+3) = (a-2)(b+3). Factor the denominator, which is a perfect square trinomial: b2+6b+9=(b+3)2b^2+6b+9 = (b+3)^2. The expression becomes (a−2)(b+3)(b+3)(b+3)\frac{(a-2)(b+3)}{(b+3)(b+3)}. Cancel one common factor of (b+3)(b+3). The simplified result is a−2b+3\frac{a-2}{b+3}.

Question 9

Simplify x3+8x2−2x+4\frac{x^3+8}{x^2-2x+4}.

  1. x−2x-2
  2. x+2x+2 (correct answer)
  3. x+4x+4
  4. The expression cannot be simplified.
Explanation: The numerator x3+8x^3+8 is a sum of cubes, x3+23x^3+2^3. The sum of cubes formula is a3+b3=(a+b)(a2−ab+b2)a^3+b^3 = (a+b)(a^2-ab+b^2). Applying this, x3+8=(x+2)(x2−2x+4)x^3+8 = (x+2)(x^2-2x+4). The expression becomes (x+2)(x2−2x+4)x2−2x+4\frac{(x+2)(x^2-2x+4)}{x^2-2x+4}. The factor (x2−2x+4)(x^2-2x+4) cancels out, leaving x+2x+2.

Question 10

For x≠3x \neq 3, what is the simplified form of the expression 6x−182x−6\frac{6x-18}{2x-6}?

  1. 3x3x
  2. x−3x-3
  3. 3x\frac{3}{x}
  4. 33 (correct answer)
Explanation: Factor the greatest common factor from the numerator and the denominator. The GCF of the numerator 6x−186x-18 is 6, so it factors to 6(x−3)6(x-3). The GCF of the denominator 2x−62x-6 is 2, so it factors to 2(x−3)2(x-3). The expression becomes 6(x−3)2(x−3)\frac{6(x-3)}{2(x-3)}. The common factor (x−3)(x-3) cancels out, leaving 62\frac{6}{2}, which simplifies to 3.

Question 11

Simplify the expression x3+x2−4x−4x2−x−2\frac{x^3+x^2-4x-4}{x^2-x-2}.

  1. x−2x-2
  2. x+1x+1
  3. x+2x+2 (correct answer)
  4. x2−4x−2\frac{x^2-4}{x-2}
Explanation: First, factor the numerator x3+x2−4x−4x^3+x^2-4x-4 by grouping. Group the first two and last two terms: x2(x+1)−4(x+1)x^2(x+1) - 4(x+1). This factors to (x2−4)(x+1)(x^2-4)(x+1). The factor (x2−4)(x^2-4) is a difference of squares, so it factors further into (x−2)(x+2)(x-2)(x+2). The fully factored numerator is (x−2)(x+2)(x+1)(x-2)(x+2)(x+1). Next, factor the denominator x2−x−2x^2-x-2 into (x−2)(x+1)(x-2)(x+1). The expression is (x−2)(x+2)(x+1)(x−2)(x+1)\frac{(x-2)(x+2)(x+1)}{(x-2)(x+1)}. Cancel the common factors (x−2)(x-2) and (x+1)(x+1). The remaining simplified expression is x+2x+2.

Question 12

Which of the following is equivalent to 25−x2x2−2x−15\frac{25-x^2}{x^2-2x-15}?

  1. x+5x+3\frac{x+5}{x+3}
  2. x−5x+3\frac{x-5}{x+3}
  3. −(x+5)x+3\frac{-(x+5)}{x+3} (correct answer)
  4. 5−xx−3\frac{5-x}{x-3}
Explanation: Factor the numerator 25−x225-x^2 as a difference of squares to get (5−x)(5+x)(5-x)(5+x). Factor the denominator x2−2x−15x^2-2x-15 to get (x−5)(x+3)(x-5)(x+3). The expression is (5−x)(5+x)(x−5)(x+3)\frac{(5-x)(5+x)}{(x-5)(x+3)}. The terms (5−x)(5-x) and (x−5)(x-5) are opposites, so (5−x)=−1(x−5)(5-x) = -1(x-5). Substitute this in: −1(x−5)(x+5)(x−5)(x+3)\frac{-1(x-5)(x+5)}{(x-5)(x+3)}. Cancel the common factor (x−5)(x-5). The simplified expression is −(x+5)x+3\frac{-(x+5)}{x+3}.

Question 13

What is the simplified form of the expression 4x2−1002x+10\frac{4x^2 - 100}{2x+10}?

  1. 2x−52x-5
  2. 2x+102x+10
  3. x−5x-5
  4. 2x−102x-10 (correct answer)
Explanation: First, factor out the greatest common factor from the numerator, which is 4: 4(x2−25)4(x^2-25). Then, factor out the GCF from the denominator, which is 2: 2(x+5)2(x+5). The expression is 4(x2−25)2(x+5)\frac{4(x^2-25)}{2(x+5)}. Factor the difference of squares in the numerator: x2−25=(x−5)(x+5)x^2-25 = (x-5)(x+5). The expression becomes 4(x−5)(x+5)2(x+5)\frac{4(x-5)(x+5)}{2(x+5)}. Cancel the common factor (x+5)(x+5) and simplify the constants 42=2\frac{4}{2}=2. The result is 2(x−5)2(x-5), which is 2x−102x-10.

Question 14

Which of the following is equivalent to the expression x2−2x−15x2−9\frac{x^2 - 2x - 15}{x^2 - 9} for all values of xx for which the expression is defined?

  1. x−5x−3\frac{x-5}{x-3} (correct answer)
  2. x+5x+3\frac{x+5}{x+3}
  3. x−5x+3\frac{x-5}{x+3}
  4. x+5x−3\frac{x+5}{x-3}
Explanation: To simplify the rational expression, factor the numerator and the denominator. The numerator x2−2x−15x^2 - 2x - 15 factors to (x−5)(x+3)(x-5)(x+3). The denominator x2−9x^2 - 9 is a difference of squares and factors to (x−3)(x+3)(x-3)(x+3). The expression becomes (x−5)(x+3)(x−3)(x+3)\frac{(x-5)(x+3)}{(x-3)(x+3)}. The common factor (x+3)(x+3) can be canceled, leaving x−5x−3\frac{x-5}{x-3}.

Question 15

Which of the following is equivalent to m3−8m2+2m+4÷m−2m+1\frac{m^3 - 8}{m^2 + 2m + 4} \div \frac{m - 2}{m + 1}?

  1. m2+2m+4m+1\frac{m^2 + 2m + 4}{m + 1}
  2. m+1m + 1 (correct answer)
  3. (m−2)(m+1)m2+2m+4\frac{(m - 2)(m + 1)}{m^2 + 2m + 4}
  4. m−2m+1\frac{m - 2}{m + 1}
Explanation: Recognize that m3−8=(m−2)(m2+2m+4)m^3 - 8 = (m-2)(m^2+2m+4) using the difference of cubes formula. The expression becomes (m−2)(m2+2m+4)m2+2m+4÷m−2m+1=(m−2)×m+1m−2=m+1\frac{(m-2)(m^2+2m+4)}{m^2+2m+4} \div \frac{m-2}{m+1} = (m-2) \times \frac{m+1}{m-2} = m+1 after canceling (m2+2m+4)(m^2+2m+4) and then (m−2)(m-2). Choice A stops after the first cancellation. Choice C inverts the division incorrectly. Choice D represents the original divisor unchanged.

Question 16

Simplify the rational expression 3x2+12xx2+x−12\frac{3x^2 + 12x}{x^2 + x - 12}.

  1. 3xx−3\frac{3x}{x-3} (correct answer)
  2. 3xx+3\frac{3x}{x+3}
  3. xx−3\frac{x}{x-3}
  4. 3x−1\frac{3}{x-1}
Explanation: First, factor the numerator by taking out the greatest common factor, 3x3x, which gives 3x(x+4)3x(x+4). Next, factor the denominator, a trinomial, which becomes (x+4)(x−3)(x+4)(x-3). The expression is now 3x(x+4)(x+4)(x−3)\frac{3x(x+4)}{(x+4)(x-3)}. Cancel the common factor (x+4)(x+4) from the numerator and denominator. The simplified expression is 3xx−3\frac{3x}{x-3}.

Question 17

Simplify the expression x4−16x2−x−2\frac{x^4 - 16}{x^2 - x - 2}.

  1. (x−2)(x2+4)(x-2)(x^2+4)
  2. (x+2)(x2+4)x+1\frac{(x+2)(x^2+4)}{x+1} (correct answer)
  3. (x−2)(x2+4)x+1\frac{(x-2)(x^2+4)}{x+1}
  4. x2+8x^2+8
Explanation: First, factor the numerator x4−16x^4 - 16 as a difference of squares: (x2−4)(x2+4)(x^2-4)(x^2+4). Then factor (x2−4)(x^2-4) further as another difference of squares: (x−2)(x+2)(x-2)(x+2). The fully factored numerator is (x−2)(x+2)(x2+4)(x-2)(x+2)(x^2+4). Next, factor the denominator x2−x−2x^2 - x - 2 into (x−2)(x+1)(x-2)(x+1). The expression becomes (x−2)(x+2)(x2+4)(x−2)(x+1)\frac{(x-2)(x+2)(x^2+4)}{(x-2)(x+1)}. Cancel the common factor (x−2)(x-2) to get (x+2)(x2+4)x+1\frac{(x+2)(x^2+4)}{x+1}.

Question 18

Simplify the expression x3−2x2+5x−10x2+5\frac{x^3 - 2x^2 + 5x - 10}{x^2+5}.

  1. x−2x-2 (correct answer)
  2. x+2x+2
  3. x−5x-5
  4. The expression cannot be simplified.
Explanation: The numerator x3−2x2+5x−10x^3 - 2x^2 + 5x - 10 can be factored by grouping. Group the first two terms and the last two terms: x2(x−2)+5(x−2)x^2(x-2) + 5(x-2). Factor out the common binomial (x−2)(x-2) to get (x2+5)(x−2)(x^2+5)(x-2). The expression is now (x2+5)(x−2)x2+5\frac{(x^2+5)(x-2)}{x^2+5}. The denominator x2+5x^2+5 is a prime polynomial but is a common factor with the numerator. Canceling (x2+5)(x^2+5) leaves x−2x-2.

Question 19

Simplify the expression 5x4−20x210x3+20x2\frac{5x^4 - 20x^2}{10x^3+20x^2}.

  1. x−22\frac{x-2}{2} (correct answer)
  2. x+22\frac{x+2}{2}
  3. x−2x+2\frac{x-2}{x+2}
  4. x2−42x+4\frac{x^2-4}{2x+4}
Explanation: First, factor the GCF from the numerator, 5x25x^2, to get 5x2(x2−4)5x^2(x^2-4). Then factor the GCF from the denominator, 10x210x^2, to get 10x2(x+2)10x^2(x+2). The expression is 5x2(x2−4)10x2(x+2)\frac{5x^2(x^2-4)}{10x^2(x+2)}. Factor the difference of squares in the numerator: 5x2(x−2)(x+2)10x2(x+2)\frac{5x^2(x-2)(x+2)}{10x^2(x+2)}. Cancel the common factor (x+2)(x+2). Also, simplify the monomial factors: 5x210x2=12\frac{5x^2}{10x^2} = \frac{1}{2}. The remaining expression is 1(x−2)2\frac{1(x-2)}{2}, or x−22\frac{x-2}{2}.

Question 20

Which expression is equivalent to x2−4x+1x2−4x+4x2−1\frac{\frac{x^2 - 4}{x + 1}}{\frac{x^2 - 4x + 4}{x^2 - 1}}?

  1. x−1x+1\frac{x - 1}{x + 1}
  2. x+2x−2\frac{x + 2}{x - 2}
  3. (x+2)(x−1)x−2\frac{(x + 2)(x - 1)}{x - 2} (correct answer)
  4. (x+2)(x+1)(x−2)2\frac{(x + 2)(x + 1)}{(x - 2)^2}
Explanation: When you encounter complex fractions like this one, remember that dividing by a fraction is the same as multiplying by its reciprocal. This problem tests your ability to factor polynomials and simplify rational expressions. To solve x2−4x+1x2−4x+4x2−1\frac{\frac{x^2 - 4}{x + 1}}{\frac{x^2 - 4x + 4}{x^2 - 1}}, first rewrite it as a multiplication: x2−4x+1×x2−1x2−4x+4\frac{x^2 - 4}{x + 1} \times \frac{x^2 - 1}{x^2 - 4x + 4} Next, factor each polynomial. The numerator x2−4x^2 - 4 is a difference of squares: (x+2)(x−2)(x + 2)(x - 2). The denominator x2−4x+4x^2 - 4x + 4 is a perfect square trinomial: (x−2)2(x - 2)^2. The expression x2−1x^2 - 1 is also a difference of squares: (x+1)(x−1)(x + 1)(x - 1). Substituting the factored forms: (x+2)(x−2)x+1×(x+1)(x−1)(x−2)2\frac{(x + 2)(x - 2)}{x + 1} \times \frac{(x + 1)(x - 1)}{(x - 2)^2} This becomes: (x+2)(x−2)(x+1)(x−1)(x+1)(x−2)2\frac{(x + 2)(x - 2)(x + 1)(x - 1)}{(x + 1)(x - 2)^2} Cancel common factors: (x+1)(x + 1) cancels completely, and one factor of (x−2)(x - 2) cancels, leaving (x+2)(x−1)x−2\frac{(x + 2)(x - 1)}{x - 2}, which is choice C. Choice A is missing the (x+2)(x + 2) factor. Choice B incorrectly has (x+2)(x + 2) in the numerator but (x−2)(x - 2) in the denominator without the (x−1)(x - 1) term. Choice D fails to cancel the common (x+1)(x + 1) factor and retains (x−2)2(x - 2)^2 in the denominator. Always factor completely before canceling, and double-check that you've simplified all possible common factors.