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.