Question

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

Answer 1

`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.