Question

Solving polynomial functions?

solve 5x^3 - 6x^2 - 4x - 8 = 0

I kinda understand polynomials, not very much, but I'm extremely confused as how to solve them. If anyone could walk me through this, that'd be great.

Answer 1

Have you been taught the rational roots theorem yet?

Answer 2

Not that I can recall

Answer 3

Okay. This is a cubic equation and therefore it should have 3 roots (or 3 zeros).
Try a few small values for x such as 1, -1, 2, -2, 3, -3 etc. and see if f(x) becomes zero. If it does you would have found one root. Because what is a root. It is the value of x that makes f(x) = 0.

Answer 4

I will do x = 1 and you can try the rest until you find one root.

5x^3 - 6x^2 - 4x - 8
Try x = 1
5(1)^3 - 6(1)^2 - 4(1) - 8
5 - 6 - 4 - 8 = -13
That is not 0 and therefore x = 1 is not a root.

Next try x = -1, then x = 2, etc. and let me know as soon as you find the first root.

Answer 5

2 is a root, because
5(2)^3 -6(2)^2 - 4(2) - 8

40 - 24 - 8 - 8 = 0

Answer 6

look at the value of constant, is -8. the factors of -8 is {+-1, +-2, +-4, +-8}. one of or more the values in parantheses can be a factor or a solution for f(x)

Answer 7

Yes! 2 is a root! Do you know synthetic division?

Answer 8

I've been told to use it but never taught how.

Answer 9

You just found that x = 2 is a root. That means (x-2) is a factor
So the original polynomial equation
5x^3 - 6x^2 - 4x - 8 = 0 can be written as:
(x-2) * (some other polynomial) = 0 because when you put x = 2,
(x-2) becomes 0 and its product will be a 0.
5x^3 - 6x^2 - 4x - 8 = (x-2) * (some other polynomial)
To find the "some other polynomial" we need to divide both sides by (x -2)

What is (5x^3 - 6x^2 - 4x - 8) / (x-2) ?

We have to do either long division or synthetic division to find the quotient.

I will do synthetic division here:

Answer 10

I'm watching some videos about synthetic division and I kind of see how it works. I will try it out myself if you want to save yourself some typing. Thank you for explaining the equation I'd use for synthetic division!

Answer 11

|
2 | 5 -6 -4 -8
| 10 8 8
|___________________________________
5 4 4 0

Answer 12

The quadratic expression will be: 5x^2 + 4x + 4

(5x^3 - 6x^2 - 4x - 8) = (x - 2)(5x^2 + 4x + 4)

See if you can factor 5x^2 + 4x + 4. If not use the quadratic formula and find the two roots.

Answer 13

you are welcome. BTW, you will get two complex roots from the quadratic. And one real root 2 that you found earlier for a total of 3 roots for the cubic equation.

Answer 14

-2/5 + 4/5i, -2/5 - 4/5i, 2

Answer 15

Yay! You got all three roots correct.