QuestionFactor. $10 y^{2}+17 y+7$

Studdy Solution

STEP 1

What is this asking? We need to rewrite this expression $10y^2 + 17y + 7$ as the product of two simpler expressions. Watch out! Don't forget to check your answer by expanding it back out; it's a super easy way to catch mistakes!

STEP 2

1. Find the factors
2. Verify

STEP 3

Alright, let's factor this quadratic expression!
We're looking for two numbers that multiply to $(10 \cdot 7) = \mathbf{70}$ and add up to $\mathbf{17}$ .
Think, think, think!

STEP 4

Eureka! Those magic numbers are 7 and 10!
See, $7 \cdot 10 = 70$ and $7 + 10 = 17$ .
Perfect!

STEP 5

Now, let's rewrite our original expression using these magic numbers: $10y^2 + 17y + 7 = 10y^2 + 10y + 7y + 7$ We've just split the middle term using our magic numbers.
Sneaky, right?

STEP 6

Time to factor by grouping!
Look at the first two terms: $10y^2 + 10y$ .
We can factor out a $10y$ , giving us $10y(y+1)$ .

STEP 7

Now, look at the last two terms: $7y + 7$ .
We can factor out a $7$ , giving us $7(y+1)$ .

STEP 8

Notice the magic!
Both terms now contain $(y+1)$ .
Let's factor that out: $10y(y+1) + 7(y+1) = (10y + 7)(y+1)$ Boom! We've factored our quadratic!

STEP 9

Let's double-check our work by expanding $(10y+7)(y+1)$ .

STEP 10

Using the FOIL method (First, Outer, Inner, Last), we get: $(10y+7)(y+1) = (10y \cdot y) + (10y \cdot 1) + (7 \cdot y) + (7 \cdot 1)$

STEP 11

Simplifying, we get: $10y^2 + 10y + 7y + 7 = 10y^2 + 17y + 7$ Woohoo! It matches our original expression.
We're factoring masters!

STEP 12

The factored form of $10y^2 + 17y + 7$ is $(10y+7)(y+1)$ .

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