Math  /  Algebra

QuestionFactor completely. 5v240v1005 v^{2}-40 v-100

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression as a product of the *simplest* possible pieces! Watch out! Don't forget to look for common factors *first*!
It makes everything way easier.

STEP 2

1. Factor out the greatest common factor.
2. Factor the quadratic expression.

STEP 3

Alright, let's **kick things off** by looking for a **greatest common factor (GCF)**!
We've got 5v240v1005v^2 - 40v - 100.
Notice that each term is divisible by **5**!
So, let's **factor that 5 out**!

STEP 4

Dividing each term by 5, we get: 5(v28v20)5(v^2 - 8v - 20) Remember, factoring out a 5 is the same as dividing each term by 5 and then multiplying the entire expression by 5.
We can check our work by distributing the 5 back in, and we should get our original expression back.

STEP 5

Now, we've got v28v20v^2 - 8v - 20 inside the parentheses.
This looks like a quadratic expression that we can **factor**!
We're looking for two numbers that **multiply** to 20-20 and **add up** to 8-8.

STEP 6

Let's think about the factors of 20-20.
We could have 1-1 and 2020, 11 and 20-20, 2-2 and 1010, 22 and 10-10, 4-4 and 55, or 44 and 5-5.
Which pair adds up to 8-8?
It's 22 and 10-10!

STEP 7

So, we can rewrite the quadratic expression as (v+2)(v10)(v + 2)(v - 10).
Don't forget that 55 we factored out earlier!

STEP 8

Putting it all together, our **completely factored expression** is: 5(v+2)(v10)5(v + 2)(v - 10) Boom! We nailed it!

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