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
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Solution

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Solution Steps

Step 1: Identify the denominators

The given product is: x24x213x2+6xx2+8xx2+11x+24 \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 3x2+6x3x^{2} + 6x and x2+11x+24x^{2} + 11x + 24.

Step 2: Factor the denominators

Factor 3x2+6x3x^{2} + 6x: 3x2+6x=3x(x+2) 3x^{2} + 6x = 3x(x + 2) Factor x2+11x+24x^{2} + 11x + 24: x2+11x+24=(x+3)(x+8) x^{2} + 11x + 24 = (x + 3)(x + 8)

Step 3: Determine the excluded values

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

  1. 3x(x+2)=03x(x + 2) = 0:
    • 3x=0x=03x = 0 \Rightarrow x = 0
    • x+2=0x=2x + 2 = 0 \Rightarrow x = -2
  2. (x+3)(x+8)=0(x + 3)(x + 8) = 0:
    • x+3=0x=3x + 3 = 0 \Rightarrow x = -3
    • x+8=0x=8x + 8 = 0 \Rightarrow x = -8

The excluded values are 00, 2-2, 3-3, and 8-8. From the given options, the excluded values are 00, 3-3, and 8-8.

Final Answer

0,3,8\boxed{0, -3, -8}

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