Questions: Simplify. (c^2+7c-18)/(c^2-9) * (c^2-5c+6)/(c^2+12c+27)=

Transcript text: Simplify. \[ \frac{c^{2}+7 c-18}{c^{2}-9} \cdot \frac{c^{2}-5 c+6}{c^{2}+12 c+27}= \]

Solution

Solution Steps

To simplify the given expression, we need to factor each polynomial in the numerators and denominators. After factoring, we can cancel out any common factors between the numerators and denominators.

Step 1: Factor the Polynomials

We start by factoring the given polynomials in the expression:

The numerator \( c^2 + 7c - 18 \) factors to \( (c - 2)(c + 9) \).
The denominator \( c^2 - 9 \) factors to \( (c - 3)(c + 3) \).
The second numerator \( c^2 - 5c + 6 \) factors to \( (c - 3)(c - 2) \).
The second denominator \( c^2 + 12c + 27 \) factors to \( (c + 3)(c + 9) \).

Step 2: Write the Expression with Factored Terms

Substituting the factored forms into the original expression, we have:

\[ \frac{(c - 2)(c + 9)}{(c - 3)(c + 3)} \cdot \frac{(c - 3)(c - 2)}{(c + 3)(c + 9)} \]

Step 3: Cancel Common Factors

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

The factor \( (c - 2) \) appears in both the first numerator and the second numerator.
The factor \( (c + 9) \) appears in both the first numerator and the second denominator.
The factor \( (c - 3) \) appears in both the second numerator and the first denominator.
The factor \( (c + 3) \) appears in both the first denominator and the second denominator.

After canceling, we are left with:

\[ \frac{(c - 2)^2}{(c + 3)^2} \]