Questions: Select all the correct answers. Consider this product. (x^2-4x-21)/(3x^2+6x) * (x^2+8x)/(x^2+11x+24) Which values are excluded values for the product? 0 2 -3 -8 7

Select all the correct answers.

Consider this product.
(x^2-4x-21)/(3x^2+6x) * (x^2+8x)/(x^2+11x+24)

Which values are excluded values for the product?
0
2
-3
-8
7

Transcript text: Select all the correct answers. Consider this product. \[ \frac{x^{2}-4 x-21}{3 x^{2}+6 x} \cdot \frac{x^{2}+8 x}{x^{2}+11 x+24} \] Which values are excluded values for the product? 0 2 -3 $-8$ 7

Solution

Solution Steps

Step 1: Identify the denominators

The given product is: \[ \frac{x^{2}-4 x-21}{3 x^{2}+6 x} \cdot \frac{x^{2}+8 x}{x^{2}+11 x+24} \] The denominators are $3x^{2} + 6x$ and $x^{2} + 11x + 24$.

Step 2: Factor the denominators

Factor $3x^{2} + 6x$: \[ 3x^{2} + 6x = 3x(x + 2) \] Factor $x^{2} + 11x + 24$: \[ x^{2} + 11x + 24 = (x + 3)(x + 8) \]

Step 3: Determine the excluded values

The excluded values are the values of $x$ that make any denominator zero. Set each factor equal to zero and solve for $x$:

$3x(x + 2) = 0$:
- $3x = 0 \Rightarrow x = 0$
- $x + 2 = 0 \Rightarrow x = -2$
$(x + 3)(x + 8) = 0$:
- $x + 3 = 0 \Rightarrow x = -3$
- $x + 8 = 0 \Rightarrow x = -8$

The excluded values are $0$, $-2$, $-3$, and $-8$. From the given options, the excluded values are $0$, $-3$, and $-8$.