Beginning Algebra With Applications, Chapter 7, 7.1, Section 7.1, Problem 36

Which sum will be zero $P + Q, Q + R,$ or $P + R$?

If we add $P$ and $Q$, we have


$
\begin{equation}
\begin{aligned}

P + Q =& (ax^3 + bx^2 - cx + d) + (-ax^3 - bx^2 + cx - d)
&&
\\
\\
=& (ax^3 - ax^3) + (bx^2- bx^2) + (-cx + cx) + (d-d)
&& \text{Use the commutative and associative properties of addition to rearrange and group like terms.}
\\
\\
=& 0
&& \text{Combine like terms}

\end{aligned}
\end{equation}
$


If we add $Q$ and $R$, we have


$
\begin{equation}
\begin{aligned}

Q + R =& (-ax^3 - bx^2 + cx - d) + (-ax^3 + bx^2 + cx + d)
&&
\\
\\
=& (-ax^3 - ax^3) + (-bx^2 + bx^2) + (cx + cx) + (-d + d)
&& \text{Use the commutative and associative properties of addition to rearrange and group like terms.}
\\
\\
=& -2ax^3 + 2cx
&& \text{ Combine like terms and write the polynomial in descending order.}

\end{aligned}
\end{equation}
$


If we add $P$ and $R$, we have


$
\begin{equation}
\begin{aligned}

P + R =& (ax^3 + bx^2 - cx + d) + (-ax^3 + bx^2 + cx + d)
\\
\\
=& (ax^3 - ax^3) + (bx^2 + bx^2) + (-cx+ cx) + (d + d)
&& \text{Use the commutative and associative properties of addition to rearrange and group like terms.}
\\
\\
=& 2bx^2 + 2d
&& \text{ Combine like terms and write the polynomial in descending order.}


\end{aligned}
\end{equation}
$


So, $P+Q$ has the sum equal to zero.