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: \[ \frac{(y + 2)}{(y - 5)} \quad \text{for } y \neq -2 \]

Step 5: State any restrictions

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

1. \(y - 5 \neq 0 \Rightarrow y \neq 5\)
2. \(y + 2 \neq 0 \Rightarrow y \neq -2\)

Final answer

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

The restrictions on the variable are:

1. \(y \neq 5, -2\)

So you would fill it in as:

1. \( y \neq -2, 5 \)

In conclusion:

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