poopey
Answered

What is the product?

[tex]\[
\frac{x^2-16}{2x+8} \cdot \frac{x^3-2x^2+x}{x^2+3x-4}
\][/tex]

A. \(\frac{x(x-4)(x-1)}{2(x+4)}\)

B. \(\frac{x(x-1)}{2}\)

C. \(\frac{(x+4)(x-4)}{2x(x-1)}\)

D. [tex]\(\frac{(x-4)(x-1)}{2x(x+4)}\)[/tex]



Answer :

Certainly! Let's solve this question step-by-step.

We need to determine the product of the following two expressions:

[tex]\[ \frac{x^2 - 16}{2x + 8} \cdot \frac{x^3 - 2x^2 + x}{x^2 + 3x - 4} \][/tex]

### Step 1: Simplify Each Expression

First, let's simplify each fraction separately.

Simplifying the first fraction:

[tex]\[ \frac{x^2 - 16}{2x + 8} \][/tex]

- Notice that \( x^2 - 16 \) is a difference of squares, which can be factored as \( (x + 4)(x - 4) \).

- Also, the denominator \( 2x + 8 \) can be factored out as \( 2(x + 4) \).

So, the first fraction can be rewritten as:

[tex]\[ \frac{(x + 4)(x - 4)}{2(x + 4)} \][/tex]

- We can cancel out the \( x + 4 \) terms in the numerator and denominator:

[tex]\[ \frac{x - 4}{2} \][/tex]

Simplifying the second fraction:

[tex]\[ \frac{x^3 - 2x^2 + x}{x^2 + 3x - 4} \][/tex]

- The numerator \( x^3 - 2x^2 + x \) can be factored by taking out the common factor \( x \):

[tex]\[ x(x^2 - 2x + 1) \][/tex]

- The quadratic \( x^2 - 2x + 1 \) is a perfect square, so it can be factored as:

[tex]\[ x(x - 1)^2 \][/tex]

Thus, the numerator becomes \( x(x - 1)^2 \).

- The denominator \( x^2 + 3x - 4 \) is the quadratic expression which can be factored into:

[tex]\[ (x + 4)(x - 1) \][/tex]

So, the second fraction can be rewritten as:

[tex]\[ \frac{x(x - 1)^2}{(x + 4)(x - 1)} \][/tex]

- We can cancel out an \( x - 1 \) term in the numerator and the denominator:

[tex]\[ \frac{x(x - 1)}{x + 4} \][/tex]

### Step 2: Calculate the Product of the Simplified Expressions

Now, we multiply the simplified forms of the two fractions:

[tex]\[ \left(\frac{x - 4}{2}\right) \cdot \left(\frac{x(x - 1)}{x + 4}\right) \][/tex]

We multiply the numerators together and the denominators together:

[tex]\[ =\frac{(x - 4) \cdot x \cdot (x - 1)}{2 \cdot (x + 4)} \][/tex]

Simplifying the expression, we get:

[tex]\[ \frac{x(x - 4)(x - 1)}{2(x + 4)} \][/tex]

This matches one of the given options exactly:

[tex]\[ \frac{x(x - 4)(x - 1)}{2(x + 4)} \][/tex]

### Step 3: Identify the Correct Answer from the Choices

[tex]\[ \boxed{\frac{x(x - 4)(x - 1)}{2(x + 4)}} \][/tex]
(-) \(\frac{x(x-1)}{2}\)

\(\frac{(x+4)(x-4)}{2 x(x-1)}\)

\(\frac{(x-4)(x-1)}{2 x(x+4)}\)

### Conclusion

By comparing the simplified product with the given choices, we find that the correct equation is:

[tex]\[ \boxed{\frac{x(x - 4)(x - 1)}{2(x + 4)}} \][/tex]

Therefore, the product is:

[tex]\[ \frac{x(x - 4)(x - 1)}{2(x + 4)} \][/tex]

The correct answer is the first choice:
[tex]\[ \boxed{\frac{x(x - 4)(x - 1)}{2(x + 4)}} \][/tex]