Question 42·Medium·Advanced Math

Type $y=\frac{x^3-9x^2+27x-27}{x-3}$. Desmos draws a parabola-shaped curve. The given $x\ne3$ matters: $x-3$ is zero there, so the fraction has no value at that input. The graph may not visibly show the single missing point, called a hole.

Test the choices on a second line, replacing each one if its curve pulls away. Type $y=(x-3)^2$. Desmos shows complete overlap with the original wherever $x\ne3$. A single crossing wouldn't establish equivalence; this curve matches throughout the allowed inputs. So the equivalent expression is $(x-3)^2$. Choice A.

A perfect-cube identity says $a^3-3a^2b+3ab^2-b^3=(a-b)^3$. Here $a=x$ and $b=3$: those values produce all four terms of the numerator. Match the terms:

Now write that pattern as a cube:

The numerator has three copies of $x-3$, while the denominator has one. Cancel one whole factor, which is allowed because the given $x\ne3$ makes it nonzero:

Cancel factors, not pieces of a sum, and keep the original exclusion. The equivalent expression is $(x-3)^2$. Choice A.