Question 100·200 Super-Hard SAT Math Questions·Advanced Math

The factor theorem says that if $x-c$ divides a polynomial, the polynomial equals $0$ at $x=c$. Here $x-5a=0$ when $x=5a$, so the choice we want must equal $0$ at $x=5a$.

Each choice has the form $6x^2-bx+35a+15$, where $b$ is the positive number multiplying $-x$. Substitute $x=5a$ and set the polynomial equal to $0$:

Square $5a$:

Divide by $5$:

Add $ab$ to both sides:

Divide by $a$, which is positive and can't be $0$:

Every choice makes $b$ an integer, and $30a+7$ is an integer when $a$ is an integer. So $\frac{3}{a}$ must also be an integer. The positive divisors of $3$ are $1$ and $3$, so those are the only values of $a$ to check.

Type $d=[1,3]$, then $q(t)=30t+7+\frac{3}{t}$, then $q(d)$. Desmos shows $[40,98]$: these are the required values of $b$ for $a=1$ and $a=3$. Only $40$ appears in a choice, so $a=1$ makes $x-5a$ a factor of $6x^2-40x+35a+15$. Choice B.