Question

Prove that (5x - 4) is a factor of the polynomial f(x) = 5x³ - 4x² - 5x +4. Hence factorize It completely.

Answer 1

If 5x - 4 is assumed to be factor, then x = ${\displaystyle \frac{4}{5}}$ . Substituting this in problem polynomial, we get:

${\displaystyle \text{f}\left(\frac{4}{5}\right)=5\times \left(\frac{4}{5}\right)\times \left(\frac{4}{5}\right)\times \left(\frac{4}{5}\right)-4\times \left(\frac{4}{5}\right)\times \left(\frac{4}{5}\right)-5\times \left(\frac{4}{5}\right)+4}$

${\displaystyle =\frac{64}{25}-\frac{64}{25}-4+4}$

= 0

Hence (5x - 4) is a factor of the polynomial. 

Multiplying (5x-4) by x², we get 5x³ - 4x², hence we are left with -5x + 4 (and 1st part of factor as x²).

Multiplying (5x - 4) by -1, we get -5x + 4, hence we are left with 0 (and 2nd part of factor as -7x). 

Hence complete factor is (5x - 4) (x²-1). 

Further factorizing (x² - 1), we get :

⇒ (x - 1)(x + 1) = 0

Hence answer is (5x - 4)(x - 1)(x + 1) = 0