How do you write a polynomial when given three roots?

## Polynomial

# 2 Answers

r_{1} = root 1 ---> **x - r _{1} = 0**

r_{2} = root 2 ----> **x - r _{2} = 0**

r_{3} = root 3 ----> **x - r _{3} = 0**

you just line them up and distribute

(x - r_{1})(x - r_{2})(x - r_{3}) = 0

and you end up with

x^{3} - **(r _{1} + r_{2} + r_{3})**x

^{2}+

**(r**x -

_{1}r_{2}+ r_{1}r_{3}+ r_{2}r_{3})**(r**= 0

_{1}*r_{2}*r_{3})

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 x^{2}-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) = (x^{2}+2x)(x-3) = x^{3}-3x^{2}+2x^{2}-6x =** x ^{3}-x^{2}-6x**

So in my sample, x

^{3}-x

^{2}-6x is the polynomial with roots -2, 0, and 3

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