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. $q - p$
b. $p + q$
c. $pq$
d. $p - q$

Explanation:

Step1: Expand polynomial Q

$$\begin{align*} Q&=(5y^2 - 4y)(3y^2 + 7)\\ &=5y^2\cdot3y^2 + 5y^2\cdot7 - 4y\cdot3y^2 - 4y\cdot7\\ &=15y^4 + 35y^2 - 12y^3 - 28y\\ &=15y^4 - 12y^3 + 35y^2 - 28y \end{align*}$$

Step2: Test Option A (Q-P)

$$\begin{align*} Q-P&=(15y^4 - 12y^3 + 35y^2 - 28y)-(8y^4 + 6y^3 + 8y)\\ &=15y^4 -12y^3 +35y^2 -28y -8y^4 -6y^3 -8y\\ &=7y^4 -18y^3 +35y^2 -36y \end{align*}$$

Step3: Test Option B (P+Q)

$$\begin{align*} P+Q&=(8y^4 + 6y^3 + 8y)+(15y^4 - 12y^3 + 35y^2 - 28y)\\ &=8y^4+15y^4 +6y^3-12y^3 +35y^2 +8y-28y\\ &=23y^4 -6y^3 +35y^2 -20y \end{align*}$$

Step4: Verify match

The result of $P+Q$ matches the target expression.

Answer:

B. $P + Q$