When the algebraic expression \frac{4x^3 - 16x}{2x^2 + 4x} is simplified for all x \neq 0 … — GRE General Test

When the algebraic expression $\frac{4x^3 - 16x}{2x^2 + 4x}$ is simplified for all $x \neq 0$ and $x \neq -2$ , it reduces to the linear polynomial $ax + b$ , where $a$ and $b$ are constants. What is the value of $a + b$ ?

Answer: -2

Answer

The correct numerical answer is -2.

Factoring the numerator yields

4x(x - 2)(x + 2)

and factoring the denominator yields

2x(x + 2)

. Canceling the common factors

2x

and

(x + 2)

leaves

2(x - 2) = 2x - 4

. Comparing

2x - 4

ax + b

gives

a = 2

and

b = -4

. Summing these values gives

a + b = 2 + (-4) = -2

Step-by-Step Solution

Factor out the greatest common factor and apply the difference of squares formula to the numerator.

4x^3 - 16x = 4x(x^2 - 4) = 4x(x - 2)(x + 2)

Fully factoring the numerator allows identification of all linear factors.

Factor out the greatest common factor from the denominator.

2x^2 + 4x = 2x(x + 2)

Extracting

2x

reveals the common terms shared with the numerator.

Divide the numerator by the denominator by canceling identical non-zero factors

2x

and

(x + 2)

\frac{4x(x - 2)(x + 2)}{2x(x + 2)} = \frac{4}{2}(x - 2) = 2(x - 2) = 2x - 4

Simplifying rational expressions requires canceling common factors present in both numerator and denominator.

Compare the simplified expression

2x - 4

to the form

ax + b

to find

a

and

b

, then compute their sum.

a = 2

and

b = -4 \implies a + b = 2 + (-4) = -2

Matching corresponding terms identifies the values of the target constants.

Key Concept

Simplifying rational expressions by factoring out common terms and applying the difference of squares identity.

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