Help on finding Real Roots of polynomials?

Question

Does anyone know of a calculator website or can help me find the roots of the polynomials

or a quick and efficient way of finding the roots, Im a bit confused and don't know where to start. I already know the quadratics but not polynomials

x^3-x^2-3x+3
6X^6-3x^5-7x^4-6x^3
x^5-2x^4+3x^+6x
x^5+4x^3+x^2+6x
x^4-t^3+2x^2-4x-8
x^5-x^3+x

Thanks
Narumi

Answer 1

You need to factorise them

x^3-x^2-3x+3
= x²(x - 1) - 3(x - 1)
= (x² - 3)(x - 1)
= 0 => x = 1 or ± √3

6x^6 - 3x^5 - 7x^4 - 6x^3
= x^3(6x^3 - 3x*2 - 7x - 6) Since f(1) < 0 and f12) > 0 there is a root of 6x^3 - 3x*2 - 7x - 6 between 1 and 2
x = 0, 0, 0 and -1.61 (approx using Newton's method ) This also has 2 complex roots

x^5-2x^4+3x^+6x?????????????? 3x^ what??
Try 3x^2
x^5 - 2x^4 +3x^2 + 6x
=x(x^4 - 2x^3 + 3x + 6)
= 0
By graphing x^4 - 2x^3 + 3x + 6 is positive definite so no more real factors.
So x = 0 (other roots are complex)

OR
Try 3x^3
x^5 - 2x^4 +3x^3 + 6x
= x(x^4 - 2x^3 + 3x2 + 6)
= 0
By graphing x^4 - 2x^3 + 3x^2 + 6 is positive definite so no more real factors
So x = 0 (other roots are complex)

x^5+4x^3+x^2+6x
= x(x^4 + 4x^2 + x + 6)
= 0
Has only 1 real root ... x = 0 (x^4 + 4x^2 + 6 > x for all x)

x^4-x^3+2x^2-4x-8 (f (-1)= 0 so x + 1 is a factor)
= (x + 1)(x^3 -2x^2 + 4x -8) (f(2) = 0 so x - 2 is a factor)
= (x + 1)(x - 2)(x^2 + 4) (x^2 + 4 will not factorise in real numbers)
= 0
x = -2. -1

x^5-x^3+x
= x(x^4 - x^2 + 1) x^4 - x^2 + 1 = u² - u + 1 where u = x² No real roots)
= 0
So real x = 0 only

Answer 2

Some of the theorems involved in solving polynomials are described on the website linked below. For polynomials with powers greater than 2, there isn't one simple technique like the quadratic formula. Rather, there are a number of techniques you can use to narrow down and test the candidates. The quickest (but not always the most accurate) way to find the roots is to use a graphing calculator. Graph the expression under y = x^3-x^2-3x+3 and find the zeros. The zeros of the polynomial function will be the roots of the polynomial expression. Hope that helps.

Answer 3

to solve all your questions in one

www.quickmath.com, just click on Factor under Algebra, then type it in just the way you have it, and if it can be factored it will factor it for you.

In case you can't find it, here's a hint, look to the left of the screen.