Let $P$ be a cubic polynomial with $P(0) = k$, $P(1) = 2k$, and $P(-1) = 3k$. What is $P(2) + P(-2)$ ?
I know that a cubic polynomial has the form $P(x) = ax^3 + bx^2 + cx + d$, where $a, b, c, d$ are constants. I also know that $P(0) = k$ means that $d = k$, since all the other terms vanish when $x = 0$. So I can write $P(x) = ax^3 + bx^2 + cx + k$. Now I need to find $a, b, c$ using the other given values. Plugging in $x = 1$, I get $P(1) = a + b + c + k = 2k$, which implies that $a + b + c = k$. Plugging in $x = -1$, I get $P(-1) = -a + b - c + k = 3k$, which implies that $-a + b - c = 2k$. Adding these two equations, I get $2b = 3k$, so $b = \frac{3}{2}k$. Substituting this into either equation, I get $a + c = -\frac{1}{2}k$. To find $a$ and $c$, I need another equation. I notice that $P(2) + P(-2)$ is the expression I want to find, so maybe I can use that somehow. Plugging in $x = 2$, I get $P(2) = 8a + 4b + 2c + k$. Plugging in $x = -2$, I get $P(-2) = -8a + 4b - 2c + k$. Adding these two equations, I get $P(2) + P(-2) = 8b + 2k = 12k + 2k = 14k$. This is the answer I want, and I didn't even need to find $a$ and $c$. 