What is (y^2 +4y+4)/(y^2 -3y-10) in simplest form? State any restrictions on the variable.

Question

The simplified form is ____.
What are the restrictions on the variable? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
1) y(=/)____ (Use a comma to separate answers as needed.)
2)There are no restrictions on y.

Answer 1

To simplify the expression $(y^2+4y+4)/(y^2-3y-10)$, we will factor both the numerator and the denominator.

Step 1: Factor the numerator

The numerator is $y^2+4y+4$. This can be factored as: $$y^2+4y+4=(y+2)(y+2)=(y+2{)}^2$$

Step 2: Factor the denominator

The denominator is $y^2-3y-10$. We need to find two numbers that multiply to $-10$ and add to $-3$. The numbers $-5$ and $2$ work. Therefore, it can be factored as: $$y^2-3y-10=(y-5)(y+2)$$

Step 3: Write the expression with the factored form

Now we can rewrite the expression: $$\frac{(y+2{)}^2}{(y-5)(y+2)}$$

Step 4: Simplify the expression

We can cancel one $(y+2)$ from the numerator and the denominator: $$\frac{(y+2)}{(y-5)}\text{for }y\ne -2$$

Step 5: State any restrictions

The original expression had restrictions based on the denominator, so we must ensure the denominator is not zero:

1. $y-5\ne 0\Rightarrow y\ne 5$
2. $y+2\ne 0\Rightarrow y\ne -2$

Final answer

The simplified form is: $$\frac{(y+2)}{(y-5)}$$

The restrictions on the variable are:

1. $y\ne 5,-2$

So you would fill it in as:

1. $y\ne -2,5$

In conclusion:

• The simplified form is $\frac{(y+2)}{(y-5)}$.
• The restrictions on $y$ are $y\ne -2,5$.