Questions: Homework #2 Multiply and simplify to lowest terms. (z^2-36)/(z^2+4z-12) * (z^2+6z+8)/(z^2-2z-24) = □

Homework #2 Multiply and simplify to lowest terms. (z^2-36)/(z^2+4z-12) * (z^2+6z+8)/(z^2-2z-24) = □
Transcript text: Homework \#2 Multiply and simplify to lowest terms. \[ \frac{z^{2}-36}{z^{2}+4 z-12} \cdot \frac{z^{2}+6 z+8}{z^{2}-2 z-24} \] $\frac{z^{2}-36}{z^{2}+4 z-12} \cdot \frac{z^{2}+6 z+8}{z^{2}-2 z-24}=$ $\square$
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Solution

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Solution Steps

To solve the given problem, we need to multiply and simplify the given rational expressions. Here are the high-level steps:

  1. Factorize all the polynomials in the numerators and denominators.
  2. Cancel out any common factors between the numerators and denominators.
  3. Multiply the remaining factors to get the simplified expression.
Step 1: Factor the Expressions

We start by factoring the given polynomials:

  1. \( z^2 - 36 = (z - 6)(z + 6) \)
  2. \( z^2 + 4z - 12 = (z - 2)(z + 6) \)
  3. \( z^2 + 6z + 8 = (z + 2)(z + 4) \)
  4. \( z^2 - 2z - 24 = (z - 6)(z + 4) \)
Step 2: Set Up the Multiplication

Now we can rewrite the original expression using the factored forms:

\[ \frac{(z - 6)(z + 6)}{(z - 2)(z + 6)} \cdot \frac{(z + 2)(z + 4)}{(z - 6)(z + 4)} \]

Step 3: Cancel Common Factors

Next, we cancel the common factors in the numerator and denominator:

  • The factor \( (z - 6) \) cancels with \( (z - 6) \).
  • The factor \( (z + 6) \) cancels with \( (z + 6) \).
  • The factor \( (z + 4) \) cancels with \( (z + 4) \).

This simplifies our expression to:

\[ \frac{(z + 2)}{(z - 2)} \]

Final Answer

Thus, the simplified expression is:

\[ \boxed{\frac{z + 2}{z - 2}} \]

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