# How do you find all the real and complex roots of #x^4 + 7x^3 + 31x^2 + 175x + 150 = 0#?

##### 1 Answer

Use the rational root theorem to find the two Real roots

#### Explanation:

#f(x) = x^4+7x^3+31x^2+175x+150#

By the rational root theorem, any rational roots of

In addition, since all of the coefficients of

So the only possible rational roots are:

#-1, -2, -3, -5, -6, -10, -15, -25, -30, -50, -75, -150#

Trying each in turn, we find:

#f(-1) = 1-7+31-175+150 = 0#

...

#f(-6) = 1296-1512+1116-1050+150 = 0#

So

#x^4+7x^3+31x^2+175x+150#

#=(x+1)(x^3+6x^2+25x+150)#

#=(x+1)(x+6)(x^2+25)#

The remaining quadratic factor has zeros

So the roots of