Question

f(x)= 2x^3-x^2-5x-2

Answer 1

I need the factorization of it

Answer 2

Do you know about the rational root theorem?

Answer 3

It says that any possible rational roots (which are intimately related to the factors) can be determined by looking at the coefficient of the term with the highest exponent value, and the coefficient of the constant term. Specifically, any possible rational root will be a factor of the constant term divided by a factor of the leading term, and either positive or negative.

2 has factors 1,2

so our possible roots are

\[\pm \frac{2}{2},\pm\frac{2}{1},\pm\frac{1}{2}\]

Any of those values which when substituted for \(x\) gives \(f(x) = 0\) is a root. Any root \(r_n\)is a factor of the form \(x-r_n\).

Once you've found a root and made the associated factor, you can divide the polynomial by that factor to get a simpler polynomial which will have the same remaining roots as the one you started with, and may be simpler to factor.

Answer 4

Thank you this helped a lot!

Answer 5

2x^3 - x^2 - 5x - 2
x^2(2x - 1) - 1(5x + 2)
x^2(2x - 1) - 1(2x - 1 + 3x + 3)
x^2(2x - 1) - 1(2x - 1) - 3x - 3
(2x - 1)(x^2 - 1) - 3x - 3
(2x - 1)(x + 1)(x - 1) - 3(x + 1)
(x + 1)((2x - 1)(x - 1) - 3)
(x + 1)(2x^2 - 3x + 1 - 3)
(x + 1)(2x^2 - 3x - 2)
(x + 1)((2x + 1)(x - 2))

Answer 6

I started looking at that approach, got a few lines in and decided it wasn't promising. I guess I should have persevered a bit longer! I don't know that I would have spotted the trick of splitting 5x+2 like you did...