Thread: find the zeros

Suppose you know that 1/3 and -2 are zeros of 3x^4-x^3-21x^2-11x+6
Find all the zeros!

Hint: 1/3 and -2 are zero of (3x-1)(x+2).

You need the Remainder Theorem.

How's your Synthetic Division? Long division?

not really good

It's time to practice.

Every time you find a zero, use it to deflate the polynomial to one degree lower. Eventually, you'll get it down to quadratic and the last two are easy.

You could also use Vieta's Formulas and get the system


and solve it simultaneously for the other roots. You should get the same quadratic (I'm bad at synthetic division too).

Hello,

Originally Posted by gumi

Suppose you know that 1/3 and -2 are zeros of 3x^4-x^3-21x^2-11x+6
Find all the zeros!

I'd do it this way :

and are zeros of

Hence

where P is a polynomial of degree 2.

We can rewrite this way :



, but we don't care.



Since the first term is , a=1 because we would have 3x*x*ax² as the dominating term.

Hence

Let's study the constant term :
-1*2*c=6

->


Thus

Now, you can develop...