Studdy Solution
STEP 1
What is this asking?
Multiply two fractions with w in them, and then simplify the result!
Watch out!
Factoring is key, and don't forget to look for common factors that can be simplified.
Also, remember dividing by zero is a no-no!
STEP 2
1. Factor the numerators and denominators.
2. Multiply the fractions.
3. Simplify the resulting fraction.
STEP 3
Let's **factor** the first numerator: 2−18w2.
We can factor out a 2, giving us 2(1−9w2).
Notice that (1−9w2) is a difference of squares!
So, we can factor it further as (1−3w)(1+3w).
This gives us 2(1−3w)(1+3w).
Awesome!
STEP 4
Now, let's **factor** the first denominator: 3w2−8w−3.
We're looking for two numbers that multiply to (3⋅−3=−9) and add up to −8.
Those numbers are −9 and 1.
So, we rewrite the expression as 3w2−9w+w−3.
Factoring by grouping gives us 3w(w−3)+1(w−3), which simplifies to (3w+1)(w−3).
STEP 5
Time to **factor** the second numerator: 3w2−7w−6.
We need two numbers that multiply to (3⋅−6=−18) and add up to −7.
Those numbers are −9 and 2.
Rewriting and factoring by grouping: 3w2−9w+2w−6=3w(w−3)+2(w−3)=(3w+2)(w−3).
Perfect!
STEP 6
Finally, let's **factor** the second denominator: 12w2+80w−28.
First, we can factor out a 4, giving us 4(3w2+20w−7).
Now, we need two numbers that multiply to (3⋅−7=−21) and add up to 20.
These numbers are 21 and −1.
Rewriting and factoring by grouping gives us 4(3w2+21w−w−7)=4(3w(w+7)−1(w+7))=4(3w−1)(w+7).
STEP 7
Let's put those factored expressions back into the fractions and **multiply** them together:
(3w+1)(w−3)2(1−3w)(1+3w)⋅4(3w−1)(w+7)(3w+2)(w−3)
When we multiply fractions, we multiply the numerators together and the denominators together:
4(3w+1)(w−3)(3w−1)(w+7)2(1−3w)(1+3w)(3w+2)(w−3)
STEP 8
Now, the fun part!
We can **simplify** by canceling out common factors.
Notice we have a (w−3) in both the numerator and denominator.
We also have a 2 in the numerator and a 4 in the denominator, which simplifies to 21.
So, we get:
2(3w+1)(3w−1)(w+7)(1−3w)(1+3w)(3w+2)
Notice that (1−3w) and (3w−1) are very similar.
We can rewrite (1−3w) as −1(3w−1).
Now we can divide the numerator and denominator by (3w−1).
2(3w+1)(3w−1)(w+7)−1(3w−1)(1+3w)(3w+2)=2(3w+1)(w+7)−1(1+3w)(3w+2)
STEP 9
Our **final simplified expression** is:
2(3w+1)(w+7)−(3w+1)(3w+2)=2(w+7)−(3w+2)
Remember that w cannot be a value that makes the denominator zero, so w=−31, w=31, w=3 and w=−7.
