Question

Factor $f(x) = x^4 - 3x^3 - 5x^2 + 21x + 22$ completely. Then sketch the graph with at least six points, including all roots and the y-intercept.

Answer 1

The rational root theorem suggests the following possible rational roots:

22, -22, 11, -11, 2, -2, 1, -1

(obtained by dividing all divisors of the constant term, 22, by the coefficient of the leading term, 1). Checking all of these possibilities, we happen to find that $f(-2)=f(-1)=0$.

By the polynomial remainder theorem (if $p(c)=0$ for some polynomial $p(x)$, then $x-c$ divides $p(x)$), we then know that $x+2$ and $x+1$ are factors.

Performing the division gives

$\dfrac{x^4-3x^3-5x^2+21x+22}{(x+2)(x+1)}=x^2-6x+11$

The remaining quadratic has discriminant

$\Delta=(-6)^2-4(1)(11)=-8<0$

which means the remaining two terms are complex and the quadratic cannot be factored any further (over the real numbers). So

$x^4 - 3 x^3 - 5 x^2 + 21 x + 22=(x+2)(x+1)(x^2-6x+11)$