Write the polynomial expression in simplest form: 6x/x−4⋅x2−16/4x+16

Question

.(1 point)
Responses

3x2
Start Fraction 3 x over 2 End Fraction

6x4
Start Fraction 6 x over 4 End Fraction

6x(x+4)4(x−4)
Start Fraction 6 x left parenthesis x plus 4 right parenthesis over 4 left parenthesis x minus 4 right parenthesis End Fraction

3x(x+4)2(x−4)

Answer 1

To simplify the expression \(\frac{6x}{x - 4} \cdot \frac{x^2 - 16}{4x + 16}\), we start by factoring where possible.

1. The expression \(x^2 - 16\) is a difference of squares, which factors as: \[ x^2 - 16 = (x - 4)(x + 4) \]

2. The expression \(4x + 16\) can factor out a common factor of 4: \[ 4x + 16 = 4(x + 4) \]

Now we can rewrite the entire expression with these factorizations: \[ \frac{6x}{x - 4} \cdot \frac{(x - 4)(x + 4)}{4(x + 4)} \]

Next, we can cancel common factors in the expression:

• The factor \(x - 4\) in the numerator and the denominator can be canceled.
• The factor \(x + 4\) also cancels.

After canceling, we have: \[ \frac{6x}{4} = \frac{3x}{2} \]

Thus, the simplest form of the polynomial expression is: \[ \frac{3x}{2} \]

This corresponds to:

Response: Start Fraction 3 x over 2 End Fraction