Question

Find the value of p for which the polynomial 3x^3 -x^2 + px +1 is exactly divisible by x-1, hence factorise the polynomial

Answer 1

By the polynomial remainder theorem, if
`3x^3-x^2+px+1` is divisible by
`x-1`, then the remainder is 0 and

`3\cdot1^3-1^2+p\cdot1+1=p+3=0\implies\boxed{p=-3}`

Now,

`3x^3-x^2-3x+1=x^2(3x-1)-(3x-1)=(x^2-1)(3x-1)=\boxed{(x-1)(x+1)(3x-1)}`