Question

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 )

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. \(P + Q\)