For what value of $k$ does $p(x)=x^3+k{x}^2-2x+5$ leave a remainder of 7 when divided by $(x-1)$?

**When $p(x)$ is divided by $(x-a)$, the remainder = $p$(a).**

Don't do long division — just substitute $x=a$ into the polynomial.

Example: divide $p(x)=x^3-2x+1$ by $(x-2)$. Remainder = $p(2)=8-4+1=5$.

Why it works: any polynomial can be written as $p(x)=q(x)\times (x-a)+r$, where $r$ is a constant remainder. Substituting $x=a$ kills the $q(x)\times (x-a)$ term, leaving $p(a)=r$.