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Difficulty: MediumSimplifying and Factoring Algebraic Expressions

For all real numbers xx such that x≠−2x \neq -2, x≠2x \neq 2, and x≠4x \neq 4, which of the following expressions are equivalent to x4−16x2x3−4x2−4x+16\frac{x^4 - 16x^2}{x^3 - 4x^2 - 4x + 16}? Select all such expressions.

  1. x3+4x2x2−4\frac{x^3 + 4x^2}{x^2 - 4}Answer
  2. x2(x+4)(x−2)(x+2)\frac{x^2(x + 4)}{(x - 2)(x + 2)}Answer
  3. C
    x2(x−4)x2−4\frac{x^2(x - 4)}{x^2 - 4}
  4. D
    x2+4xx−2\frac{x^2 + 4x}{x - 2}
  5. E
    x4−4x2x3−4\frac{x^4 - 4x^2}{x^3 - 4}

Answer

The equivalent expressions are x3+4x2x2−4\frac{x^3 + 4x^2}{x^2 - 4} and x2(x+4)(x−2)(x+2)\frac{x^2(x + 4)}{(x - 2)(x + 2)}.
The given expression factors as x2(x−4)(x+4)(x2−4)(x−4)\frac{x^2(x - 4)(x + 4)}{(x^2 - 4)(x - 4)}. Canceling (x−4)(x - 4) leaves x2(x+4)x2−4\frac{x^2(x + 4)}{x^2 - 4}. Expanding the numerator gives x3+4x2x2−4\frac{x^3 + 4x^2}{x^2 - 4}, and factoring the denominator further gives x2(x+4)(x−2)(x+2)\frac{x^2(x + 4)}{(x - 2)(x + 2)}. Both represent valid equivalent forms of the expression.

Step-by-Step Solution

1
Factor the numerator of the given algebraic expression.
x4−16x2=x2(x2−16)=x2(x−4)(x+4)x^4 - 16x^2 = x^2(x^2 - 16) = x^2(x - 4)(x + 4)
Factor out the greatest common factor x2x^2, then apply the difference of squares identity a2−b2=(a−b)(a+b)a^2 - b^2 = (a - b)(a + b).
2
Factor the denominator by grouping terms.
x3−4x2−4x+16=x2(x−4)−4(x−4)=(x2−4)(x−4)=(x−2)(x+2)(x−4)x^3 - 4x^2 - 4x + 16 = x^2(x - 4) - 4(x - 4) = (x^2 - 4)(x - 4) = (x - 2)(x + 2)(x - 4)
Group the first two terms and last two terms, factor out common binomials, and expand the remaining difference of squares.
3
Simplify the full rational expression by canceling common factors.
x2(x−4)(x+4)(x2−4)(x−4)=x2(x+4)x2−4=x3+4x2x2−4\frac{x^2(x - 4)(x + 4)}{(x^2 - 4)(x - 4)} = \frac{x^2(x + 4)}{x^2 - 4} = \frac{x^3 + 4x^2}{x^2 - 4}
Since x≠4x \neq 4, cancel the common factor (x−4)(x - 4) from both numerator and denominator.
4
Compare the simplified form to the given choices to identify all equivalent expressions.
Both x3+4x2x2−4\frac{x^3 + 4x^2}{x^2 - 4} and x2(x+4)(x−2)(x+2)\frac{x^2(x + 4)}{(x - 2)(x + 2)} match the simplified algebraic forms.
Expanding the numerator or factoring the denominator yields these two equivalent representations.

Key Concept

Simplifying rational algebraic expressions by polynomial factoring and term grouping.
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Topics

Simplifying and Factoring Algebraic Expressions

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