Question

consider the expression \\(-3(y-5)^2-9+7y\\).
which statements are true about the process and simplified product? check all that apply.

the first step in simplifying is to distribute the \\(-3\\) throughout the parentheses.
there are 3 terms in the simplified product.
the simplified product is a degree 3 polynomial.
the final simplified product is \\(-3y^2+7y-9\\).
the final simplified product is \\(-3y^2+37y-84\\).

Explanation:

Expand the binomial square

Using the Perfect Square Trinomial knowledge point

$$ (y - 5)^2 = y^2 - 10y + 25 $$

Distribute the constant factor

Using the Distributive Property knowledge point

$$ -3(y^2 - 10y + 25) = -3y^2 + 30y - 75 $$

Combine like terms

Using the Like Terms knowledge point

$$
LATEXBLOCK0
$$

Analyze the simplified polynomial characteristics

Using the Polynomial Degree and Polynomial Classification knowledge points

• The simplified expression \(-3y^2 + 37y - 84\) has a degree of 2.
• It contains exactly 3 terms: \(-3y^2\), \(37y\), and \(-84\).

Evaluate the given statements

• Statement 1: "The first step in simplifying is to distribute the \(-3\) throughout the parentheses."
• False: Order of operations requires evaluating the exponent \((y-5)^2\) before multiplying by \(-3\).
• Statement 2: "There are 3 terms in the simplified product."
• True: The terms are \(-3y^2\), \(37y\), and \(-84\).
• Statement 3: "The simplified product is a degree 3 polynomial."
• False: The highest exponent is 2, so it is a degree 2 polynomial.
• Statement 4: "The final simplified product is \(-3y^2 + 7y - 9\)."
• False.
• Statement 5: "The final simplified product is \(-3y^2 + 37y - 84\)."
• True.

Answer:

• The first step in simplifying is to distribute the \(-3\) throughout the parentheses.
• There are 3 terms in the simplified product. (Correct answer)
• The simplified product is a degree 3 polynomial.
• The final simplified product is \(-3y^2 + 7y - 9\).
• The final simplified product is \(-3y^2 + 37y - 84\). (Correct answer)