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Difficulty: MediumEquivalent Algebraic Expressions

Which of the following is equivalent to the expression 2x2−7x−4x2−16⋅x2+4x2x+1\frac{2x^2 - 7x - 4}{x^2 - 16} \cdot \frac{x^2 + 4x}{2x + 1} for all values of xx where the expression is defined?

  1. A
    x(2x−1)2x+1\frac{x(2x-1)}{2x+1}
  2. xxAnswer
  3. C
    x2x^2
  4. D
    x(x+4)x−4\frac{x(x+4)}{x-4}

Answer

xx
The correct answer is xx because factoring all numerators and denominators of the given product results in (2x+1)(x−4)(x−4)(x+4)⋅x(x+4)2x+1\frac{(2x + 1)(x - 4)}{(x - 4)(x + 4)} \cdot \frac{x(x + 4)}{2x + 1}. After cancelling the common factors of (2x+1)(2x + 1), (x−4)(x - 4), and (x+4)(x + 4), only xx remains.

Step-by-Step Solution

1
Factor the quadratic expression in the numerator of the first fraction.
2x2−7x−4=(2x+1)(x−4)2x^2 - 7x - 4 = (2x + 1)(x - 4)
Finding the factors of the quadratic trinomial allows us to look for common terms to cancel later.
2
Factor the denominator of the first fraction using the difference of squares identity.
x2−16=(x−4)(x+4)x^2 - 16 = (x - 4)(x + 4)
Simplifying the quadratic binomial in the denominator exposes common factors.
3
Factor the numerator of the second fraction by extracting the greatest common factor.
x2+4x=x(x+4)x^2 + 4x = x(x + 4)
Exhibiting the shared variable xx helps identify terms that will cancel with the denominator.
4
Substitute the factored forms back into the original expression and cancel the common factors.
(2x+1)(x−4)(x−4)(x+4)⋅x(x+4)2x+1=x\frac{(2x + 1)(x - 4)}{(x - 4)(x + 4)} \cdot \frac{x(x + 4)}{2x + 1} = x
The terms (2x+1)(2x + 1), (x−4)(x - 4), and (x+4)(x + 4) appear in both the numerator and the denominator, leaving only the factor xx.

Key Concept

Simplifying rational expressions by factoring and cancelling common factors
Estimated Time:1m 30s
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Topics

Equivalent Algebraic Expressions

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