Question

Given f(x) = 3x^3+ kx – 11, and x – 1 is a factor of f(x), then what is the value
of k?

Answer 1

let's recall the remainder theorem.

we know that (x-1) is a factor, that means x -1 = 0 or x = 1.

since we know that (x-1) is a factor, then dividing the polynomial by it will give us a remainder of 0, which correlates with saying that f(1) = 0, in this case, so we can simply plug in "1" as the argument, knowing it gives 0.

$f(x)=3x^3+kx-11\\\\[-0.35em]~\dotfill\\\\\stackrel{0}{f(1)}=3(1)^3+k(1)-11\implies \stackrel{f(1)}{0}=3+k-11\implies 0=-8+k\implies 8=k$