Question

Prove that  (x+ 1) is a factor of x³ - 6x² + 5x + 12 and hence factorize it completely.

Answer 1

If x + 1 is assumed to be factor, then x= -1. Substituting this in problem polynomial, we get: 

f( -1) = (-1) × (-1) × (-1) - 6 × (-1) × (-1) + 5 × (-1) + 12 = 0

Hence ( x + 1) is a factcr of the polynomial.  

Multiplying (x + 1) by x², we get x³ + x², hence we are left with -7x² + 5x + 12 (and 1st part of factor as x²). 

Multiplying (x + 1) by -7x, we get -7x² - 7x, hence we are left with 12x + 12 (and 2nd part of factor as -7x).

Multiplying (x +1) by 12, we get 12x + 12, hence we are left with 0 (and 3rd part of factor as 12).

Hence complete factor is (x+1) (x²-7x+12). 

Further factorizing (x² - 7x + 12), we get: 

x² - 3x - 4x + 12 =O 

⇒ (x - 4)(x - 3) = 0 

Hence answer is (x + 1)(x - 4)(x - 3) = 0