Write the polynomial expression in simplest form: 2x^2−32/x−4 ⋅ 10x+20/5x^2−20

Question

A. 4
B. 4(x+4)/x−2
C. 20(x+4)5(x−2)
D. 4(x+4)/x+2

Answer 1

To simplify the expression \( \frac{2x^2 - 32}{x - 4} \cdot \frac{10x + 20}{5x^2 - 20} \), we'll start by factoring where possible.

1. Factor \( 2x^2 - 32 \): \[ 2x^2 - 32 = 2(x^2 - 16) = 2(x - 4)(x + 4) \]

2. Factor \( 10x + 20 \): \[ 10x + 20 = 10(x + 2) \]

3. Factor \( 5x^2 - 20 \): \[ 5x^2 - 20 = 5(x^2 - 4) = 5(x - 2)(x + 2) \]

Now, we can rewrite the original expression using these factors: \[ \frac{2(x - 4)(x + 4)}{x - 4} \cdot \frac{10(x + 2)}{5(x - 2)(x + 2)} \]

Next, simplify the expression:

• The \( x - 4 \) in the numerator and denominator cancels out: \[ = 2(x + 4) \cdot \frac{10(x + 2)}{5(x - 2)(x + 2)} \]

The \( x + 2 \) in the numerator and denominator also cancels out: \[ = 2(x + 4) \cdot \frac{10}{5(x - 2)} \]

Now, simplify \( \frac{10}{5} = 2 \): \[ = 2(x + 4) \cdot \frac{2}{(x - 2)} = \frac{4(x + 4)}{(x - 2)} \]

Thus, the simplest form of the polynomial expression is: \[ \frac{4(x + 4)}{x - 2} \]

The correct answer is: B. \( \frac{4(x+4)}{x-2} \).