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)#