Math / Algebra

Question $\begin{array}{l}y^{2}+2 y-24 y^{2}+8 y+16 \\ \frac{y^{2}+12 y+32}{y^{2}+2 y-24} \cdot \frac{y^{2}+5 y-36}{y^{2}+8 y+16}=\end{array}$

Studdy Solution

STEP 1

What is this asking? Multiply these two fractions with $y$ s, then simplify the result! Watch out! Factoring is key, and don't forget to simplify only after multiplying.

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: $y^2 + 12y + 32$ .
We're looking for two numbers that add to $\boldsymbol{12}$ and multiply to $\boldsymbol{32}$ .
Those numbers are $\boldsymbol{4}$ and $\boldsymbol{8}$ !
So, $y^2 + 12y + 32 = (y+4)(y+8)$ .
Awesome!

STEP 4

Now, let's factor the first denominator: $y^2 + 2y - 24$ .
We need two numbers that add to $\boldsymbol{2}$ and multiply to $\boldsymbol{-24}$ .
That's $\boldsymbol{6}$ and $\boldsymbol{-4}$ !
So, $y^2 + 2y - 24 = (y+6)(y-4)$ .
Fantastic!

STEP 5

Time to factor the second numerator: $y^2 + 5y - 36$ .
We're searching for two numbers that add to $\boldsymbol{5}$ and multiply to $\boldsymbol{-36}$ .
That's $\boldsymbol{9}$ and $\boldsymbol{-4}$ !
So, $y^2 + 5y - 36 = (y+9)(y-4)$ .
Keep it up!

STEP 6

Finally, let's factor the second denominator: $y^2 + 8y + 16$ .
We need two numbers that add to $\boldsymbol{8}$ and multiply to $\boldsymbol{16}$ .
That's $\boldsymbol{4}$ and $\boldsymbol{4}$ !
So, $y^2 + 8y + 16 = (y+4)(y+4)$ .
We're on a roll!

STEP 7

Now, we multiply the factored fractions: $\frac{(y+4)(y+8)}{(y+6)(y-4)} \cdot \frac{(y+9)(y-4)}{(y+4)(y+4)}$

STEP 8

This becomes: $\frac{(y+4)(y+8)(y+9)(y-4)}{(y+6)(y-4)(y+4)(y+4)}$

STEP 9

Notice we have a $(y+4)$ and a $(y-4)$ in both the numerator and the denominator.
We can divide both the numerator and denominator by $(y+4)$ , which is the same as multiplying by $\frac{1/(y+4)}{1/(y+4)} = 1$ .
We can also divide both by $(y-4)$ , which is the same as multiplying by $\frac{1/(y-4)}{1/(y-4)} = 1$ .
Since we're multiplying by $1$ , we're not changing the value of the expression.

STEP 10

After simplifying, we get: $\frac{(y+8)(y+9)}{(y+6)(y+4)}$

STEP 11

Our final simplified answer is: $\frac{(y+8)(y+9)}{(y+6)(y+4)}$ .

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Question $\begin{array}{l}y^{2}+2 y-24 y^{2}+8 y+16 \\ \frac{y^{2}+12 y+32}{y^{2}+2 y-24} \cdot \frac{y^{2}+5 y-36}{y^{2}+8 y+16}=\end{array}$

Studdy Solution

STEP 1

What is this asking? Multiply these two fractions with $y$ s, then simplify the result! Watch out! Factoring is key, and don't forget to simplify only after multiplying.

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: $y^2 + 12y + 32$ .
We're looking for two numbers that add to $\boldsymbol{12}$ and multiply to $\boldsymbol{32}$ .
Those numbers are $\boldsymbol{4}$ and $\boldsymbol{8}$ !
So, $y^2 + 12y + 32 = (y+4)(y+8)$ .
Awesome!

STEP 4

Now, let's factor the first denominator: $y^2 + 2y - 24$ .
We need two numbers that add to $\boldsymbol{2}$ and multiply to $\boldsymbol{-24}$ .
That's $\boldsymbol{6}$ and $\boldsymbol{-4}$ !
So, $y^2 + 2y - 24 = (y+6)(y-4)$ .
Fantastic!

STEP 5

Time to factor the second numerator: $y^2 + 5y - 36$ .
We're searching for two numbers that add to $\boldsymbol{5}$ and multiply to $\boldsymbol{-36}$ .
That's $\boldsymbol{9}$ and $\boldsymbol{-4}$ !
So, $y^2 + 5y - 36 = (y+9)(y-4)$ .
Keep it up!

STEP 6

Finally, let's factor the second denominator: $y^2 + 8y + 16$ .
We need two numbers that add to $\boldsymbol{8}$ and multiply to $\boldsymbol{16}$ .
That's $\boldsymbol{4}$ and $\boldsymbol{4}$ !
So, $y^2 + 8y + 16 = (y+4)(y+4)$ .
We're on a roll!

STEP 7

Now, we multiply the factored fractions: $\frac{(y+4)(y+8)}{(y+6)(y-4)} \cdot \frac{(y+9)(y-4)}{(y+4)(y+4)}$

STEP 8

This becomes: $\frac{(y+4)(y+8)(y+9)(y-4)}{(y+6)(y-4)(y+4)(y+4)}$

STEP 9

STEP 10

After simplifying, we get: $\frac{(y+8)(y+9)}{(y+6)(y+4)}$

STEP 11

Our final simplified answer is: $\frac{(y+8)(y+9)}{(y+6)(y+4)}$ .

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Questiony2+2y−24y2+8y+16y2+12y+32y2+2y−24⋅y2+5y−36y2+8y+16=\begin{array}{l}y^{2}+2 y-24 y^{2}+8 y+16 \\ \frac{y^{2}+12 y+32}{y^{2}+2 y-24} \cdot \frac{y^{2}+5 y-36}{y^{2}+8 y+16}=\end{array}y2+2y−24y2+8y+16y2+2y−24y2+12y+32​⋅y2+8y+16y2+5y−36​=​

Studdy Solution

STEP 1

What is this asking? Multiply these two fractions with yyys, then simplify the result! Watch out! Factoring is key, and don't forget to simplify only *after* multiplying.

STEP 2

1. Factor the numerators and denominators.2. Multiply the fractions.3. Simplify the resulting fraction.

STEP 3

STEP 4

STEP 5

STEP 6

STEP 7

Now, we **multiply** the factored fractions: (y+4)(y+8)(y+6)(y−4)⋅(y+9)(y−4)(y+4)(y+4) \frac{(y+4)(y+8)}{(y+6)(y-4)} \cdot \frac{(y+9)(y-4)}{(y+4)(y+4)} (y+6)(y−4)(y+4)(y+8)​⋅(y+4)(y+4)(y+9)(y−4)​

STEP 8

This becomes: (y+4)(y+8)(y+9)(y−4)(y+6)(y−4)(y+4)(y+4) \frac{(y+4)(y+8)(y+9)(y-4)}{(y+6)(y-4)(y+4)(y+4)} (y+6)(y−4)(y+4)(y+4)(y+4)(y+8)(y+9)(y−4)​

STEP 9

STEP 10

After simplifying, we get: (y+8)(y+9)(y+6)(y+4) \frac{(y+8)(y+9)}{(y+6)(y+4)} (y+6)(y+4)(y+8)(y+9)​

STEP 11

Our **final simplified answer** is: (y+8)(y+9)(y+6)(y+4)\frac{(y+8)(y+9)}{(y+6)(y+4)}(y+6)(y+4)(y+8)(y+9)​.

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Start learning now

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

Question $\begin{array}{l}y^{2}+2 y-24 y^{2}+8 y+16 \\ \frac{y^{2}+12 y+32}{y^{2}+2 y-24} \cdot \frac{y^{2}+5 y-36}{y^{2}+8 y+16}=\end{array}$

What is this asking? Multiply these two fractions with $y$ s, then simplify the result! Watch out! Factoring is key, and don't forget to simplify only after multiplying.

1. Factor the numerators and denominators.
2. Multiply the fractions.
3. Simplify the resulting fraction.

Now, we multiply the factored fractions: $\frac{(y+4)(y+8)}{(y+6)(y-4)} \cdot \frac{(y+9)(y-4)}{(y+4)(y+4)}$

This becomes: $\frac{(y+4)(y+8)(y+9)(y-4)}{(y+6)(y-4)(y+4)(y+4)}$

After simplifying, we get: $\frac{(y+8)(y+9)}{(y+6)(y+4)}$

Our final simplified answer is: $\frac{(y+8)(y+9)}{(y+6)(y+4)}$ .