# How do you find all the real and complex roots of #2x^4 - 5x^3 + 53x^2 - 125x + 75 = 0#?

##### 1 Answer

Check the sum of the coefficients to find root

#### Explanation:

First note that the sum of the coefficients is zero.

That is:

So

Use your favourite method to divide

Then factor by grouping...

#f(x) = (x-1)(2x^3-3x^2+50x-75)#

#=(x-1)((2x^3-3x^2)+(50x-75))#

#=(x-1)(x^2(2x-3)+25(2x-3))#

#=(x-1)(x^2+25)(2x-3)#

The remaining quadratic factor has no linear factors with Real coefficients since

#=(x-1)(x-5i)(x+5i)(2x-3)#

So