Consider the expression $-3(y-5)^2-9+7 y$. Which statements are true about the process and simplified product? Check all that apply.

A. The first step in simplifying is to distribute the -3 throughout the parentheses.
B. There are 3 terms in the simplified product.
C. The simplified product is a degree 3 polynomial.
D. The final simplified product is $-3 y^2+7 y-9$.
E. The final simplified product is $-3 y^2+37 y-84$.

First, expand the squared term: ( y − 5 ) 2 = y 2 − 10 y + 25 .
Then, multiply by -3: − 3 ( y 2 − 10 y + 25 ) = − 3 y 2 + 30 y − 75 .
Next, add the remaining terms and combine like terms: − 3 y 2 + 30 y − 75 − 9 + 7 y = − 3 y 2 + 37 y − 84 .
Finally, determine that the correct statements are: there are 3 terms in the simplified product, and the final simplified product is − 3 y 2 + 37 y − 84 . − 3 y 2 + 37 y − 84 ​

Explanation

Analyzing the Problem We are given the expression − 3 ( y − 5 ) 2 − 9 + 7 y and asked to determine which statements about the simplification process and the simplified product are true. We will expand and simplify the expression, then check each statement.

Expanding the Square First, we expand the squared term ( y − 5 ) 2 . This gives us ( y − 5 ) ( y − 5 ) = y 2 − 5 y − 5 y + 25 = y 2 − 10 y + 25 .

Multiplying by -3 Next, we multiply the expanded term by -3: − 3 ( y 2 − 10 y + 25 ) = − 3 y 2 + 30 y − 75 .

Adding Remaining Terms Now, we add the remaining terms: − 3 y 2 + 30 y − 75 − 9 + 7 y .

Combining Like Terms Finally, we combine like terms: − 3 y 2 + ( 30 y + 7 y ) + ( − 75 − 9 ) = − 3 y 2 + 37 y − 84 .

Checking Statements Now we check each statement:

The first step in simplifying is to distribute the -3 throughout the parentheses. This is incorrect because we need to expand the square first due to order of operations (PEMDAS/BODMAS).

There are 3 terms in the simplified product. The simplified product is − 3 y 2 + 37 y − 84 , which has 3 terms.

The simplified product is a degree 3 polynomial. The simplified product is − 3 y 2 + 37 y − 84 , which is a degree 2 polynomial.

The final simplified product is − 3 y 2 + 7 y − 9 . This is incorrect.

The final simplified product is − 3 y 2 + 37 y − 84 . This is correct.


Examples
Polynomials are used to model various real-world phenomena, such as the trajectory of a ball, the growth of a population, or the behavior of electrical circuits. Simplifying polynomial expressions allows us to analyze these models more easily and make predictions about the systems they represent. For example, if we have a polynomial that represents the profit of a company, we can simplify it to find the break-even points or the maximum profit.

Answered by GinnyAnswer | 2025-07-07

In simplifying the expression − 3 ( y − 5 ) 2 − 9 + 7 y , we first expand and distribute, ultimately combining like terms to get − 3 y 2 + 37 y − 84 . The true statements are B (there are 3 terms) and E (the final product is − 3 y 2 + 37 y − 84 ). A, C, and D are not correct based on the simplification steps taken.
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Answered by Anonymous | 2025-07-22