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 = `4/5` . Substituting this in problem polynomial, we get:

`"f"(4/5) = 5 xx (4/5) xx (4/5) xx (4/5) - 4 xx (4/5) xx (4/5) - 5 xx (4/5) + 4`

`= 64/25 - 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