Question

What is the HCF of the polynomials (x – 2) (x2 + x – 2) and (x – 1) (x2

– 3x + 2)?​

Answer 1

$\underline{\textsf{Given:}}$

$\mathsf{(x-2)(x^2+x-2)}\;\textsf{and}\;\mathsf{(x-1)(x^2-3x+2)}$

$\underline{\textsf{To find:}}$

$\textsf{H.C.F of the given two polynomials}$

$\underline{\textsf{Solution:}}$

$\textsf{Highest common factor:}$

$\textsf{The Highest common factor of given two polynomials is}$

$\textsf{a common polynomial of highest degree}$

$\textsf{which divides both the polynomials}$

$\textsf{First we factorize the given polynomials}$

$\mathsf{(x-2)(x^2+x-2)=(x-2)(x+2)(x-1)}$

$\mathsf{(x-1)(x^2-3x+2)=(x-1)(x-2)(x-1)}$

$\textsf{Clearly, (x-1) and (x-2) are common factors}$

$\textsf{H.C.F=(x-1)(x-2)}$

$\mathsf{H.C.F=x^2-3x+2}$

$\underline{\textsf{Answer:}}$

$\textsf{H.C.F of the given two polynomials is}$

$\mathsf{x^2-3x+2}$

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Answer 2

$\displaystyle\huge\red{\underline{\underline{Solution}}}$

HCF OF POLYNOMIALS

HCF of two polynomials is defined as a polynomial of highest degree dividing each one of the given polynomials.

GIVEN

The given two polynomials are

$\sf{ (x - 2)( {x}^{2} + x - 2) \: \: \: and \: \: \: (x - 1)( {x}^{2} - 3x + 2) \: }$

TO DETERMINE

The HCF of the polynomials

CALCULATION

FIRST POLYNOMIAL

$\sf{ (x - 2)( {x}^{2} + x - 2)}$

$= \sf{ (x - 2)( {x}^{2} + 2x - x - 2)}$

$= \sf{ (x - 2) \{x(x + 2) -1( x + 2) \}}$

$= \sf{ (x - 2)( x- 1)(x + 2)}$

SECOND POLYNOMIAL

$\sf{(x - 1)( {x}^{2} - 3x + 2) }$

$= \sf{(x - 1)( {x}^{2} - 2x - x + 2) }$

$= \sf{(x - 1) \{x(x - 2) - 1(x - 2) \}}$

$= \sf{(x - 1)(x - 1) (x - 2)}$

RESULT

The required HCF of two polynomials

$= \sf{(x - 1)( x - 2) }$

$= \sf{ {x}^{2} - 3x + 2 }$

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