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College Algebra Quiz

College Algebra Quiz: Multiply And Divide Rational Expressions

Practice Multiply And Divide Rational Expressions in College Algebra with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 4

0 of 4 answered

If f(x)=x2−4x+3x2−9f(x) = \frac{x^2-4x+3}{x^2-9}f(x)=x2−9x2−4x+3​ and g(x)=x2+x−12x2−2x−3g(x) = \frac{x^2+x-12}{x^2-2x-3}g(x)=x2−2x−3x2+x−12​, what is f(x)⋅g(x)f(x) \cdot g(x)f(x)⋅g(x) in simplest form?

Select an answer to continue

What this quiz covers

This quiz focuses on Multiply And Divide Rational Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for College Algebra.

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

If f(x)=x2−4x+3x2−9f(x) = \frac{x^2-4x+3}{x^2-9}f(x)=x2−9x2−4x+3​ and g(x)=x2+x−12x2−2x−3g(x) = \frac{x^2+x-12}{x^2-2x-3}g(x)=x2−2x−3x2+x−12​, what is f(x)⋅g(x)f(x) \cdot g(x)f(x)⋅g(x) in simplest form?

  1. (x−1)(x+4)(x+3)(x+1)\frac{(x-1)(x+4)}{(x+3)(x+1)}(x+3)(x+1)(x−1)(x+4)​ (correct answer)
  2. x−1x+3\frac{x-1}{x+3}x+3x−1​
  3. x+4x+1\frac{x+4}{x+1}x+1x+4​
  4. (x−3)(x+4)(x+3)(x+1)\frac{(x-3)(x+4)}{(x+3)(x+1)}(x+3)(x+1)(x−3)(x+4)​

Explanation: First factor each polynomial: x2−4x+3=(x−1)(x−3)x^2-4x+3 = (x-1)(x-3)x2−4x+3=(x−1)(x−3), x2−9=(x−3)(x+3)x^2-9 = (x-3)(x+3)x2−9=(x−3)(x+3), x2+x−12=(x+4)(x−3)x^2+x-12 = (x+4)(x-3)x2+x−12=(x+4)(x−3), and x2−2x−3=(x−3)(x+1)x^2-2x-3 = (x-3)(x+1)x2−2x−3=(x−3)(x+1). So f(x)=(x−1)(x−3)(x−3)(x+3)=x−1x+3f(x) = \frac{(x-1)(x-3)}{(x-3)(x+3)} = \frac{x-1}{x+3}f(x)=(x−3)(x+3)(x−1)(x−3)​=x+3x−1​ and g(x)=(x+4)(x−3)(x−3)(x+1)=x+4x+1g(x) = \frac{(x+4)(x-3)}{(x-3)(x+1)} = \frac{x+4}{x+1}g(x)=(x−3)(x+1)(x+4)(x−3)​=x+1x+4​. Therefore, f(x)⋅g(x)=x−1x+3⋅x+4x+1=(x−1)(x+4)(x+3)(x+1)f(x) \cdot g(x) = \frac{x-1}{x+3} \cdot \frac{x+4}{x+1} = \frac{(x-1)(x+4)}{(x+3)(x+1)}f(x)⋅g(x)=x+3x−1​⋅x+1x+4​=(x+3)(x+1)(x−1)(x+4)​.

Question 2

Which expression represents x2+5x+6x2−x−2÷x2+4x+3x2−4\frac{x^2+5x+6}{x^2-x-2} \div \frac{x^2+4x+3}{x^2-4}x2−x−2x2+5x+6​÷x2−4x2+4x+3​ in lowest terms?

  1. (x+2)(x−2)(x+1)2\frac{(x+2)(x-2)}{(x+1)^2}(x+1)2(x+2)(x−2)​
  2. x+2x+1\frac{x+2}{x+1}x+1x+2​
  3. x−2x+1\frac{x-2}{x+1}x+1x−2​ (correct answer)
  4. (x+3)(x−2)(x+1)(x−1)\frac{(x+3)(x-2)}{(x+1)(x-1)}(x+1)(x−1)(x+3)(x−2)​

Explanation: Convert to multiplication: x2+5x+6x2−x−2⋅x2−4x2+4x+3\frac{x^2+5x+6}{x^2-x-2} \cdot \frac{x^2-4}{x^2+4x+3}x2−x−2x2+5x+6​⋅x2+4x+3x2−4​. Factor each polynomial: x2+5x+6=(x+2)(x+3)x^2+5x+6 = (x+2)(x+3)x2+5x+6=(x+2)(x+3), x2−x−2=(x−2)(x+1)x^2-x-2 = (x-2)(x+1)x2−x−2=(x−2)(x+1), x2−4=(x−2)(x+2)x^2-4 = (x-2)(x+2)x2−4=(x−2)(x+2), and x2+4x+3=(x+1)(x+3)x^2+4x+3 = (x+1)(x+3)x2+4x+3=(x+1)(x+3). The expression becomes (x+2)(x+3)(x−2)(x+1)⋅(x−2)(x+2)(x+1)(x+3)\frac{(x+2)(x+3)}{(x-2)(x+1)} \cdot \frac{(x-2)(x+2)}{(x+1)(x+3)}(x−2)(x+1)(x+2)(x+3)​⋅(x+1)(x+3)(x−2)(x+2)​. Cancel (x+2)(x+2)(x+2), (x+3)(x+3)(x+3), and (x+1)(x+1)(x+1) to get x−2x+1\frac{x-2}{x+1}x+1x−2​.

Question 3

Which expression is equivalent to x2+3x−10x2−25⋅x2−10x+25x2−4x+4\frac{x^2+3x-10}{x^2-25} \cdot \frac{x^2-10x+25}{x^2-4x+4}x2−25x2+3x−10​⋅x2−4x+4x2−10x+25​?

  1. x−5x−2\frac{x-5}{x-2}x−2x−5​ (correct answer)
  2. x+2x+5\frac{x+2}{x+5}x+5x+2​
  3. (x+2)(x−5)(x+5)(x−2)\frac{(x+2)(x-5)}{(x+5)(x-2)}(x+5)(x−2)(x+2)(x−5)​
  4. x+5x−2\frac{x+5}{x-2}x−2x+5​

Explanation: Factor each polynomial: x2+3x−10=(x+5)(x−2)x^2+3x-10 = (x+5)(x-2)x2+3x−10=(x+5)(x−2), x2−25=(x−5)(x+5)x^2-25 = (x-5)(x+5)x2−25=(x−5)(x+5), x2−10x+25=(x−5)2x^2-10x+25 = (x-5)^2x2−10x+25=(x−5)2, and x2−4x+4=(x−2)2x^2-4x+4 = (x-2)^2x2−4x+4=(x−2)2. The expression becomes (x+5)(x−2)(x−5)(x+5)⋅(x−5)2(x−2)2\frac{(x+5)(x-2)}{(x-5)(x+5)} \cdot \frac{(x-5)^2}{(x-2)^2}(x−5)(x+5)(x+5)(x−2)​⋅(x−2)2(x−5)2​. Cancel (x+5)(x+5)(x+5) and one factor of (x−2)(x-2)(x−2) and one factor of (x−5)(x-5)(x−5) to get x−5x−2\frac{x-5}{x-2}x−2x−5​.

Question 4

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

  1. (x−2)(x+2)2(x+2)(x−2)\frac{(x-2)(x+2)^2}{(x+2)(x-2)}(x+2)(x−2)(x−2)(x+2)2​
  2. x+2x−2\frac{x+2}{x-2}x−2x+2​
  3. x−2x+2\frac{x-2}{x+2}x+2x−2​
  4. x+2x+2x+2 (correct answer)

Explanation: Convert division to multiplication: x3−8x2+2x+4⋅x2+4x+4x2−4\frac{x^3-8}{x^2+2x+4} \cdot \frac{x^2+4x+4}{x^2-4}x2+2x+4x3−8​⋅x2−4x2+4x+4​. Factor using difference of cubes and other patterns: x3−8=(x−2)(x2+2x+4)x^3-8 = (x-2)(x^2+2x+4)x3−8=(x−2)(x2+2x+4), x2+4x+4=(x+2)2x^2+4x+4 = (x+2)^2x2+4x+4=(x+2)2, and x2−4=(x−2)(x+2)x^2-4 = (x-2)(x+2)x2−4=(x−2)(x+2). The expression becomes (x−2)(x2+2x+4)x2+2x+4⋅(x+2)2(x−2)(x+2)\frac{(x-2)(x^2+2x+4)}{x^2+2x+4} \cdot \frac{(x+2)^2}{(x-2)(x+2)}x2+2x+4(x−2)(x2+2x+4)​⋅(x−2)(x+2)(x+2)2​. Cancel (x2+2x+4)(x^2+2x+4)(x2+2x+4), (x−2)(x-2)(x−2), and one factor of (x+2)(x+2)(x+2) to get x+2x+2x+2.