Question 149·200 Super-Hard SAT Math Questions·Advanced Math

The left denominator contains all three factors. For $x\ne2,-2$, multiply both sides by $(x-2)(x+2)(x^2+1)$. Multiply each numerator by the denominator factors it is missing. Keep the minus sign on the entire final product:

Type the right side of that polynomial equation as $f(x)$ in Desmos. This gives the polynomial whose coefficients you need; you don't have to expand its products by hand.

Type $x_1=[-3,-1,0,1,3]$. Desmos stores these five distinct values as a list. None is $2$ or $-2$, so every input is allowed in the original equation.

Type the regression below: $\sim$ tells Desmos to fit the unknown letters. Use $q$ for the problem's $e$, since Desmos reserves $e$ for a number. Under PARAMETERS, Desmos shows $a=-2$, $b=5$, and $c=21$. These are the coefficients of the identity: two polynomials of degree at most $4$ that agree at five distinct inputs must be the same polynomial.

Type $a+b+c$. Desmos uses the fitted values and prints $24$. The question asks for $a+b+c$, not all five coefficients, so the value is $24$. Choice A.

In the cleared numerator, only the subtracted final product can make an $x^4$ term: $-(2x)(x)(x^2)=-2x^4$. The other products reach only $x^3$, so $a=-2$.

The first two products contribute $3x^3$ and $5x^3$. In the final product, $(2x+7)(x-2)$ has an $x$ term of $7x-4x=3x$; multiplying it by $x^2$ and applying the outside minus gives $-3x^3$. Add the $x^3$ coefficients:

The first two products contribute $-x^2$ and $10x^2$. For the final product, $(2x+7)(x-2)=2x^2+3x-14$. Multiplication by $x^2+1$ makes its $x^2$ coefficient $-14+2=-12$; the outside minus changes that to $+12$. Add:

Type $-2+5+21$ in Desmos. It prints $24$, the value of $a+b+c$. Choice A.