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