Polynomial

Question

How do you write a polynomial when given three roots?

Answer 1

r₁ = root 1 ---> x - r₁ = 0

r₂ = root 2 ----> x - r₂ = 0

r₃ = root 3 ----> x - r₃ = 0

you just line them up and distribute

(x - r₁)(x - r₂)(x - r₃) = 0

and you end up with

x³ - (r₁ + r₂ + r₃)x² + (r₁r₂ + r₁r₃ + r₂r₃)x - (r₁*r₂*r₃) = 0

 

Answer 2

Remember that the roots satisfy f(x)=0.  Some time ago you solved polynomials by factoring, and used the zero-product property to find the roots.  For example, suppose you factored x²-4x-5=0 and got (x+1)(x-5)=0.  By the zero-product property, your roots are -1 and 5.

Going from the roots to the polynomial is like going backwards.  Suppose you are given -2, 3, and 0 as roots.  You could write it in the factored form:

x(x+2)(x-3)=0

(You can check by plugging in the roots and seeing that the result=0.) 

Now to find the polynomial, multiply the terms.

x(x+2)(x-3) = (x²+2x)(x-3) = x³-3x²+2x²-6x = x³-x²-6x  
So in my sample, x³-x²-6x is the polynomial with roots -2, 0, and 3

I hope this answers your question!  Let me know if you need clarification.