How do you find all the zeros of #f(x) = x^4 – x^3 + 7x^2 – 9x – 18#?

Question

1616 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-05-30T09:15:54

#p_4(x)=(x+1)(x-2)(x+3i)(x-3i)#

The constant coefficient divided by the maximum power coefficient in a polynomial, gives the product of all its roots.

So #18/1=x_1 x_2 x_3 x_4# for a polynomial represented as

#p_4(x)=(x-x_1)(x-x_2)(x-x_3)(x-x_4)#

Supposing integer roots and testing for #pm 1,pm2,pm3,pm6#
we obtain:

#p_4(-1)=p_4(2)=0# so we have at least a root for each #1# and #-1#
then we can do

#p_4(x)=(x+1)(x-2)(x-x_3)(x-x_4)=x^4-x^3+7x^2-9x-18#

Now dividing #(x^4-x^3+7x^2-9x-18)/((x+1)(x-2)) = x^2+9#

so all the roots are

#p_4(x)=(x+1)(x-2)(x+3i)(x-3i)#