Question 9·25 Super-Hard SAT Advanced Math Questions·Advanced Math

A composition feeds one function’s output into another. Put the whole input $(x-2)^2-1$ into each factor of $f$:

Each bracket begins with an $x^2$ term, so their product begins with $x^6$. Its leading coefficient is $1$; a polynomial with that property is called monic. The given six distinct x-intercepts are six different roots, or x-values where $g(x)=0$.

Call the six roots $r_1,\ldots,r_6$. Each root $r_i$ gives a factor $x-r_i$, and monic means there’s no extra multiplier. At $x=0$:

The six minus signs cancel. For a monic polynomial with an even number of roots, its value at zero is the product of its roots.

Type $f(x)=(x-1)(x-4)(x-9)$, then $g(x)=f((x-2)^2-1)$, then $g(0)$. Desmos prints $12$. Since $g(0)$ equals the product of the six x-intercept coordinates, that product is $12$. Choice B.

Let $u=(x-2)^2-1$, the input to $f$. A product is zero when one of its factors is zero, so $f(u)=0$ when $u=1$, $4$, or $9$. Each of those inner outputs can lead to x-intercepts of $g$.

Let $t$ stand for any one of $1$, $4$, or $9$. Since $u=t$, write $(x-2)^2-1=t$. Add $1$ to both sides:

Both a positive number and its negative have the same square, so keep both signs when you take square roots:

Add $2$ to get the two x-coordinates for this value of $t$:

They form a pair centered at $2$, rather than a pair centered at $0$.

The two values are $2-\sqrt{t+1}$ and $2+\sqrt{t+1}$. Their product uses the difference of squares: $(a-b)(a+b)=a^2-b^2$. So multiply the pair:

Simplify the result:

Don’t multiply only the squared distances $t+1$: the x-coordinates include the shift by $2$.

For $t=1$, $4$, and $9$, the pair products $3-t$ are $2$, $-1$, and $-6$. Type $(3-1)(3-4)(3-9)$ in Desmos; it prints $12$. That multiplies the x-coordinates from all three pairs, so the product of all six is $12$. Choice B.