Let's consider the given expression:
[tex]\[ -3(y-5)^2 - 9 + 7y \][/tex]
We will follow a step-by-step process to simplify this expression and then check the statements provided.
1. First step in simplifying:
- Expand [tex]\( (y-5)^2 \)[/tex]:
[tex]\[
(y-5)^2 = y^2 - 10y + 25
\][/tex]
- Now distribute the [tex]\(-3\)[/tex] throughout the parentheses:
[tex]\[
-3(y^2 - 10y + 25) = -3y^2 + 30y - 75
\][/tex]
- Thus, the expression becomes:
[tex]\[
-3y^2 + 30y - 75 - 9 + 7y
\][/tex]
2. Combine like terms:
- Combine the constants [tex]\(-75\)[/tex] and [tex]\(-9\)[/tex]:
[tex]\[
-75 - 9 = -84
\][/tex]
- Combine the like terms involving [tex]\(y\)[/tex]:
[tex]\[
30y + 7y = 37y
\][/tex]
- So the expression now becomes:
[tex]\[
-3y^2 + 37y - 84
\][/tex]
We have simplified the original expression to:
[tex]\[ -3y^2 + 37y - 84 \][/tex]
Now let's determine the truth of each statement:
1. The first step in simplifying is to distribute the [tex]\(-3\)[/tex] throughout the parentheses.
- True. This is indeed the first step after squaring the binomial [tex]\((y-5)^2\)[/tex].
2. There are 3 terms in the simplified product.
- True. The simplified expression [tex]\(-3y^2 + 37y - 84\)[/tex] contains three terms: [tex]\(-3y^2, 37y,\)[/tex] and [tex]\(-84\)[/tex].
3. The simplified product is a degree 3 polynomial.
- False. The highest degree of the polynomial in the simplified expression is 2, so it is a degree 2 polynomial.
4. The final simplified product is [tex]\( -3y^2 + 7y - 9 \)[/tex].
- False. The correct simplified expression is [tex]\(-3y^2 + 37y - 84\)[/tex].
5. The final simplified product is [tex]\( -3y^2 + 37y - 84 \)[/tex].
- True. This matches our simplified expression exactly.
Thus, the valid statements are:
- The first step in simplifying is to distribute the [tex]\(-3\)[/tex] throughout the parentheses.
- There are 3 terms in the simplified product.
- The final simplified product is [tex]\( -3y^2 + 37y - 84 \)[/tex].
