Question 6·25 Super-Hard SAT Advanced Math Questions·Advanced Math

Call the numerator $g(x)=ax^5+bx^4+cx^3+dx^2+ex+f$. At $1$, every power is $1$. At $-1$, odd powers are $-1$ and even powers are $1$:

Subtracting cancels the even-power terms. Subtract $g(-1)$ from $g(1)$:

Adding the values instead would cancel the coefficients you need.

Divide by $2$ to isolate the requested sum:

The denominator on the left is $(x-2)^2(x+3)$. Multiply both sides by it to get the numerator at every allowed input:

Type this as $g(x)$ in Desmos. The original fractions rule out $x=2$ and $x=-3$, but neither is an input you'll use.

Type $g(1)$. Desmos prints $-33$, the numerator's value at $1$. That's the sum of all six coefficients, so don't stop there.

Type $g(-1)$. Desmos prints $61$. This is the second value needed for the subtraction; $-1$ is allowed because it makes neither denominator factor zero.

Type $\frac{g(1)-g(-1)}{2}$. Desmos prints $-47$. The order matters: subtract the value at $-1$ from the value at $1$. So $a+c+e=-47$. Grid in -47.