Factor 4x^3 - 9x^2 + 6x + 1?

1 Answer
May 12, 2017

#f(x)=4x^3-9x^2+6x+1# does not have rational factors.

Explanation:

If #(x-a)# is a factor of #f(x)=4x^3-9x^2+6x+1#, #a# could be a factor of #+-1/4# - here #1# comes from constant term and #4# comes from the coefficient of highest power #x^3#.

Hence, #a# could be #+-1/4#, #+-1/2# or #+-1#.

Further from factor theorem if #(x-a)# is a factor of #f(x)# then #f(a)=0#.

We know here that #f(1)=2# and #f(-1)=-18# and hence #(x-1)# and #(x+1)# are not the factors of #f(x)#. Similarly

#f(1/2)=4/8-9/4+3+1!=0# and #f(-1/2)=-4/8+9/4-3+1!=0#

#f(1/4)=4/64-9/16+3/4+1!=0# and#f(-1/4)=-4/64-9/16-3/4+1!=0#

Hence #f(x)=4x^3-9x^2+6x+1# does not have rational factors.