Question

find the hcf of polynomial x3-4x2+x+6and (6x2+x-1) (x-3)3

Answer 1

Given polynomials x³ - 4x² + x + 6 and (6x² + x - 1)(x - 1)³

Factorisation :- x³ - 4x² + x + 6
= x³ + x² - 5x² - 5x + 6x + 6
= x² (x + 1) - 5x(x + 1) + 6(x + 1)
= (x² - 5x + 6)(x + 1)
= (x² - 2x - 3x + 6)(x + 1)
= (x - 2)(x - 3)(x + 1)

Factorisation :- (6x² + x - 1)(x - 1)³
= (6x² + 3x -2x - 1)(x - 1)³
= (2x +1)(3x - 1)(x - 1)³

Common factor = 1
Hence, HCF of given polynomials is 1

Answer 2

Solution:-

given by polynomial :-

${x}^{3} - 4 {x}^{2} + x + 6 \\ {x}^{3} + {x}^{2} - 5 {x}^{2} - 5x + 6x + 6 \\ {x}^{2} ( x + 1) - 5x(x + 1) + 6(x + 1) \\ (x + 1)( {x}^{2} - 5x + 6) \\ (x + 1)( {x}^{2} - 3x - 2x + 6) \\ (x + 1)(x(x - 3) - 2(x - 3)) \\ factors \: are \: (x - 3)(x - 2)(x + 1)$

$(6 {x}^{2} + x - 1)( {x - 3)}^{3} \\ (6 {x}^{2} + 3x - 2x - 1) {(x - 3)}^{3} \\ (3x(2x + 1) - 1(2x + 1))( {(x - 3)}^{3} \\ factors \: are \: (2x + 1)(3x + 1)(x + 3)(x + 3)(x + 3)$
common factor is 1.

HCF of given polynomials is 1. ans

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