Let's analyze the expression [tex]\(-3(y-5)^2-9+7y\)[/tex] in terms of algebraic simplification and the various statements.
### Simplifying the Expression
Given:
[tex]\[ -3(y-5)^2 - 9 + 7y \][/tex]
1. Distribute the -3 Throughout the Parentheses:
To simplify the expression, we first address the term [tex]\(-3(y-5)^2\)[/tex]. The term [tex]\((y-5)^2\)[/tex] can be expanded as:
[tex]\[ (y-5)^2 = y^2 - 10y + 25 \][/tex]
Now distribute [tex]\(-3\)[/tex] to each term within the expanded form:
[tex]\[ -3(y^2 - 10y + 25) = -3y^2 + 30y - 75 \][/tex]
So, the expression becomes:
[tex]\[ -3y^2 + 30y - 75 - 9 + 7y \][/tex]
2. Combine Like Terms:
Let's combine the [tex]\(y\)[/tex] terms and the constant terms:
[tex]\(-3y^2\)[/tex] remains as it is.
[tex]\(30y + 7y = 37y\)[/tex].
[tex]\(-75 - 9 = -84\)[/tex].
Therefore, the simplified expression is:
[tex]\[ -3y^2 + 37y - 84 \][/tex]
Now, let's evaluate the given statements based on the simplification:
### Statements Analysis
1. The first step in simplifying is to distribute the -3 throughout the parentheses:
This statement is asking if the step to distribute [tex]\(-3\)[/tex] is correctly identified as the first simplification strategy. Knowing that [tex]\(-3(y-5)^2\)[/tex] should be dealt with by addressing the square first (expanding [tex]\((y-5)^2\)[/tex]) and then distributing [tex]\(-3\)[/tex], this is not accurately representing the initial necessary steps.
Thus, this statement is FALSE.
2. There are 3 terms in the simplified product:
Looking at the final simplified expression:
[tex]\[ -3y^2 + 37y - 84 \][/tex]
We see there are indeed three terms.
This statement is TRUE.
3. The simplified product is a degree 3 polynomial:
The degree of a polynomial is determined by the highest power of the variable in the simplified form. In our simplified expression:
[tex]\[ -3y^2 + 37y - 84 \][/tex]
The highest power is [tex]\(y^2\)[/tex], making this a degree 2 polynomial, not degree 3.
This statement is FALSE.
4. The final simplified product is [tex]\(-3y^2 + 7y - 9\)[/tex]:
Comparing this proposed simplified form with our actual simplified form:
[tex]\[ -3y^2 + 37y - 84 \][/tex]
The forms do not match.
Thus, this statement is FALSE.
5. The final simplified product is [tex]\(-3y^2 + 37y - 84\)[/tex]:
This matches our final simplified expression perfectly.
Thus, this statement is TRUE.
### Conclusion
From our analysis, the correct evaluations are:
- The first step in simplifying is to distribute the -3 throughout the parentheses: FALSE
- There are 3 terms in the simplified product: TRUE
- The simplified product is a degree 3 polynomial: FALSE
- The final simplified product is [tex]\(-3y^2 + 7y - 9\)[/tex]: FALSE
- The final simplified product is [tex]\(-3y^2 + 37y - 84\)[/tex]: TRUE
Thus the true statements are:
- There are 3 terms in the simplified product.
- The final simplified product is [tex]\(-3y^2 + 37y - 84\)[/tex].
