Studdy AI Logo
Math

Question Multiply the given rational expressions and simplify the result, excluding the values p=−2,3,6p = -2, 3, 6.

Studdy Solution

STEP 1

Assumptions
1. We are multiplying two rational expressions.
2. We need to factor both the numerators and denominators where possible.
3. We need to simplify the expression by canceling out common factors.
4. The variable pp cannot take the values −2-2, 33, or 66 as these would make the denominators zero.

STEP 2

First, we factor the numerator of the first fraction and the denominator of the second fraction.
p2−4p−126p+12⋅18pp2−9p+18\frac{p^2 - 4p - 12}{6p + 12} \cdot \frac{18p}{p^2 - 9p + 18}
The numerator p2−4p−12p^2 - 4p - 12 can be factored into (p−6)(p+2)(p - 6)(p + 2).

STEP 3

Now, we factor the denominator of the second fraction p2−9p+18p^2 - 9p + 18.
We look for two numbers that multiply to 1818 and add up to −9-9. These numbers are −6-6 and −3-3. So, the factored form is (p−6)(p−3)(p - 6)(p - 3).

STEP 4

We rewrite the original expression with the factored forms.
(p−6)(p+2)6p+12⋅18p(p−6)(p−3)\frac{(p - 6)(p + 2)}{6p + 12} \cdot \frac{18p}{(p - 6)(p - 3)}

STEP 5

Next, we factor the denominator of the first fraction 6p+126p + 12.
We can factor out a 66 to get 6(p+2)6(p + 2).

STEP 6

Now, we rewrite the expression with all factored forms.
(p−6)(p+2)6(p+2)⋅18p(p−6)(p−3)\frac{(p - 6)(p + 2)}{6(p + 2)} \cdot \frac{18p}{(p - 6)(p - 3)}

STEP 7

Before multiplying, we cancel out the common factors from the numerators and denominators.
The (p+2)(p + 2) terms cancel each other, and one (p−6)(p - 6) term cancels out.

STEP 8

We simplify the expression after canceling.
16⋅18pp−3\frac{1}{6} \cdot \frac{18p}{p - 3}

STEP 9

Now, we multiply the remaining terms.
16⋅18p=3p\frac{1}{6} \cdot 18p = 3p

STEP 10

The simplified product is:
3pp−3\frac{3p}{p - 3}
This is the product in simplest form, given the restrictions on pp.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord