zero of polynomials?

Question

what are the zeros of this polynomial?

x^3 - 5x^2 + 2x + 8

i know that one of them is 2 but i dont know what the other two are!

Answer 1

Hello

We can solve this several different ways.

If we have a calculator then we can graph and use the "zero" function to find the values are 4, 2, -1.

If you don't have a calculator then we can use division to find the other zeros.

If an equation has a zero then you will have factors with no remainders when you divide them.

Using the Remainder Theorem -- we can try and guess for one of the zeros and then divide.

Lets see if 0, 1, or 2 is a zero.
Find f(x) and if you get zero --- then its a factor of f(x)=x^3 - 5x^2 + 2x + 8.

So f(0) = 8 --- thus 0 is not a factor and not a zero
f(1) = 6 - not a zero
f(2) = 0 --- YES - it is a zero and a factor.

So we know that x^3 - 5x^2 + 2x + 8 = (x-2)(something)

Lets do long division:
x^3 - 5x^2 + 2x + 8 / (x-2) = x^2-3x-4.

So we have (x^2 - 3x - 4)(x-2).

Also x^2 - 3x - 4 = (x-4)(x+1)

Thus the factors are (x-4)(x+1)(x-2).
Zeros are then 4, 2, -1

Hope this helps

Answer 2

Perform long division or synthetic division to find out what
(x^3 - 5x^2 + 2x + 8)/(x-2) is. You will finf it is x^2-3x-4. This in turn = (x-4)(x+1). So your other zeroes are x = 4 and -1.

Answer 3

-1,0 2,0 4,0