# What are the possible rational roots of #x^4-5x^3+9x^2-7x+2=0# and then determine the rational roots?

##### 1 Answer

The "possible" rational roots are:

The actual roots are:

#### Explanation:

Given:

#f(x) = x^4-5x^3+9x^2-7x+2#

By the rational roots theorem, any rational zeros of

That means that the only possible rational zeros are:

#+-1, +-2#

Note also that the pattern of signs of the coefficients is:

So the only possible rational zeros are

We find:

#f(1) = 1-5+9-7+2 = 0#

So

#x^4-5x^3+9x^2-7x+2 = (x-1)(x^3-4x^2+5x-2)#

Note that the sum of the coefficients of the remaining cubic factor is also zero, so

#x^3-4x^2+5x-2 = (x-1)(x^2-3x+2)#

Note that the remaining quadratic also has

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