How do you factor the polynomials?

Question

f(x)=x^4+x^3-7x^2-13x-6. and find all of the roots of f?

g(x)=x^5+5x^4+10x^3+10x^2+5x+1 & roots of g?

thanks!

Answer 1

There really isn't any way to do this other than guess and check. If you find that you substitute some value in and find f(a) = 0, then (x-a) will be a factor. You can repeat that a few times until you have found some roots, and then do a long division to something which you *can* solve by hand (a quadratic or linear).
Note also that you only need to check things which are factors of the constant term, and there aren't too many.
In the first case, you will find that substituting in -2, -1 and 3 will all give 0, but you may have trouble finding the last root. So you will have to do some long divisions (divide by (x+2)(x+1)(x-3)), and you will end up finding that the -1 was a repeated root - so the answer is (x+2)(x+1)^2(x-3).

Same thing for the second: you will find -1 as one root. You can then do a long division and find -1 is repeated, and again, and again, and again. The actual factorisation is (x+1)^5 - you may have noticed this if you knew much about pascals triangle and recognised the coefficients are an entire row - 1,5,10,10,5,1.

Answer 2

Assuming the roots are integers: Look for factors of the constant term.

For the second polynomial with constant term 1, the only possibilities are +1 and +1, and -1 and -1. You can see quickly the +1 is not going to work because g(1) is positive for every term.

For the first polynomial, the possibilities are +6 and -1, -6 and +1, +3 and -2, -2 and +3.

If one of these works, use synthetic division with the factor that works and reduce to a lower degree polynomial, and try again.

Answer 3

The second is easier because its only possible rational roots are 1 and -1, and 1 can't work since all are plus, so keep dividing by x-1 until the end. You should be able to do this five times.

The first could have +-6, +-3, +-2, +-1 as roots. Try them until you get one that works, then divide by x - that root until you get it down to a trinomial that you can factor by FOIL or quadratic formula. To start, -1 works, so (x+1) is a factor. Divide and the quotient is x^3 - 7x - 6