# How do you find all the real and complex roots of f(x)=x^3-5x^2+11x-15?

Mar 19, 2018

${x}_{1} = 3$, ${x}_{2} = 1 - 2 i$ and ${x}_{3} = 1 + 2 i$

#### Explanation:

${x}^{3} - 5 {x}^{2} + 11 x - 15 = 0$

${x}^{3} - 3 {x}^{2} - 2 {x}^{2} + 6 x + 5 x - 15 = 0$

${x}^{2} \cdot \left(x - 3\right) - 2 x \cdot \left(x - 3\right) + 5 \cdot \left(x - 3\right) = 0$

$\left(x - 3\right) \cdot \left({x}^{2} - 2 x + 5\right) = 0$

From first multiplier, ${x}_{1} = 3$. From second one ${x}_{2} = 1 - 2 i$ and ${x}_{3} = 1 + 2 i$