Factorising Polynomial for Control Engineering Problem

[ineedmunchies]

Homework Statement

As part of a control engineering problem, I need to factorise this polynomial.

s^3 + 16s^2 + 85s + 250

(EDIT: the images failed so had to write it in this form)

Homework Equations

I know that the roots are -10, -3+4j and -3-3j.

The Attempt at a Solution

I'm not quite sure where to begin with this. Should i guess a factor and attempt long division to see if it is one? or what should i do?

 

[HallsofIvy]

You KNOW the roots. Surely you know that if a, b, and c are roots of a cubic (with leading coefficient 1) then it factors as (x- a)(x- b)(x- c).

However, the zeros of your polynomial are NOT -3+4j, -3-3j because complex zeros of a polynomial equation with real coefficents must be conjugates. -3+4j is a solution so the other must be 3- 4j, not -3-3j. That tells you immediately that the factors are (x-(-10))(x-(3+4j))((x-(3-4j)). If you want only real factors then multiply oujt the last two: (x-(3+4j))(x-(3-4j))= ((x-3)- 4j)((x-3)+4j)= (x-3)²+ 16= x²- 6x+9+ 16= x²- 6x+ 25.

x^{3+ 16x2+ 85x+ 250 factors as (x+ 10)(x2- 6x+ 25) over the real numbers and as (x+ 10)(x-3-4j)(x-3+4j) over the complex numbers.}

 

[ineedmunchies]

Yea sorry that was a typo for the first roots. shortly after writing this i realized the roots would be part of the factors and worked it out. thanks very much!