select the correct answer.
consider polynomials ( p ) and ( q ).
( p = 8y^4 + 6y^3 + 8y )
( q = (5y^2 - 4y)(3y^2 + 7) )
which operation results in an expression equivalent to ( 23y^4 - 6y^3 + 35y^2 - 20y )?
a. ( pq )
b. ( p - q )
c. ( p + q )
d. ( q - p )

Question

Accepted Answer

# Explanation:
## Step1: Expand polynomial Q
First, expand $Q=(5y^2 - 4y)(3y^2 + 7)$ using the distributive property:
$$
\begin{align*}
Q&=5y^2(3y^2 + 7) - 4y(3y^2 + 7)\
&=15y^4 + 35y^2 - 12y^3 - 28y\
&=15y^4 - 12y^3 + 35y^2 - 28y
\end{align*}
$$
## Step2: Test Option A (PQ)
The degree of $P$ is 4, degree of $Q$ is 4, so $PQ$ has degree 8, which does not match the target degree 4. Eliminate A.
## Step3: Test Option B (P - Q)
Substitute $P=8y^4 + 6y^3 + 8y$ and expanded $Q$:
$$
\begin{align*}
P-Q&=(8y^4 + 6y^3 + 8y)-(15y^4 - 12y^3 + 35y^2 - 28y)\
&=8y^4 + 6y^3 + 8y -15y^4 +12y^3 -35y^2 +28y\
&=-7y^4 +18y^3 -35y^2 +36y
\end{align*}
$$
This does not match the target expression. Eliminate B.
## Step4: Test Option C (P + Q)
Add $P$ and expanded $Q$:
$$
\begin{align*}
P+Q&=(8y^4 + 6y^3 + 8y)+(15y^4 - 12y^3 + 35y^2 - 28y)\
&=8y^4+15y^4 +6y^3-12y^3 +35y^2 +8y-28y\
&=23y^4 -6y^3 +35y^2 -20y
\end{align*}
$$
This matches the target expression.
## Step5: Verify Option D (Q-P)
$$
\begin{align*}
Q-P&=(15y^4 - 12y^3 + 35y^2 - 28y)-(8y^4 + 6y^3 + 8y)\
&=7y^4 -18y^3 +35y^2 -36y
\end{align*}
$$
This does not match the target expression. Eliminate D.

# Answer:
C. $P + Q$

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QUESTION IMAGE

Question

Explanation:

Step1: Expand polynomial Q

Step2: Test Option A (PQ)

Step3: Test Option B (P - Q)

Step4: Test Option C (P + Q)

Step5: Verify Option D (Q-P)

Answer: