Question

x³ - 5x² + 11x - 15

Explanation:

Step1: Test rational root candidates

Using Rational Root Theorem, test $x=3$:
$3^3 - 5(3)^2 + 11(3) - 15 = 27 - 45 + 33 - 15 = 0$
So $(x-3)$ is a factor.

Step2: Perform polynomial division

Divide $x^3 -5x^2 +11x -15$ by $(x-3)$:
$\frac{x^3 -5x^2 +11x -15}{x-3} = x^2 -2x +5$

Step3: Check quadratic for real roots

Discriminant of $x^2 -2x +5$:
$\Delta = (-2)^2 - 4(1)(5) = 4 - 20 = -16 < 0$
No real linear factors, so factor over reals stops here.

Answer:

$(x-3)(x^2 - 2x + 5)$