Question

Find HCF of (4x³ + 3x²y - 9xy² + 2y³) and (x² + xy - 2y²)
(1) (x + 2y)(x - y)
(2) (x - 2y)(x + y)
(3) (x - 2y)
(4) None of these

[RRB Kolkata, 2003]​

Answer 1

Answer:

HCF : Highest Common Factor of the given terms, in simple words those terms which are common factor of all the given terms.

So we will Factorise both terms.

Factorisation of First Term :

⇒ (4x³ + 3x²y - 9xy² + 2y³)

⇒ 4x³ + (7 - 4)x²y - (7 + 2)xy² + 2y³

⇒ 4x³ + 7x²y - 4x²y - 7xy² - 2xy² + 2y³

⇒ x²(4x + 7y) - xy(4x + 7y) - 2y²(x - y)

⇒ (x² - xy)(4x + 7y) - 2y²(x - y)

⇒ x(x - y)(4x + 7y) - 2y²(x - y)

⇒ (x - y)[x(4x + 7y) - 2y²]

⇒ (x - y)[4x² + 7xy - 2y²]

⇒ (x - y)[4x² + (8 - 1)xy - 2y²]

⇒ (x - y)[4x² + 8xy - xy - 2y²]

⇒ (x - y)[4x(x + 2y) - y(x + 2y)]

⇒ (x - y)(x + 2y)(4x - y)

Factorisation of Second Term :

⇒ (x² + xy - 2y²)

⇒ x² + (2 - 1)xy - 2y²

⇒ x² + 2xy - xy - 2y²

⇒ x(x + 2y) - y(x + 2y)

⇒ (x - y)(x + 2y)

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• HCF of the given terms :

⇢ HCF = Common Factor

• Common factor in both terms are (x - y) and (x + 2y)

⇢ HCF = (x + 2y)(x - y)

∴ Correct option is (1) (x + 2y)(x - y).

Answer 2

AnswEr-:

• $\underline {\boxed{\mathrm { The\:common\:\:factord\:in\:both\:terms \:are\:-:\bf{\red{(x-y)(x+2y)}}}}}$

• $\underline {\boxed{\mathrm { The\:Option \bf{\red{1}}\:or\:\bf{\red{(x-y)(x+2y)}\:is\:correct}}}}$

Explanation-:

• H.C.F or ( Highest Common Factor ) -: We have to find H.C.F of the given terms or the Common Factor of the Given terms .

There are two terms -:

1. (4x³ + 3x²y - 9xy² + 2y³)
2. (x² + xy - 2y²)

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• Factorising first term -:

• $\mathrm {First \:Term\:-: \:( 4x^{3} + 3x^{2}y - 9xy^{2} + 2y^{3})}$

• $:\implies{\mathrm { \:( 4x^{3} + 3x^{2}y - 9xy^{2} + 2y^{3})}}$

• $\sf{\dag{ By\:Using \:Sum-\:Product \:pattern-:}}$

• $:\implies{\mathrm { \: 4x^{3} + \purple {3x^{2}y} - \purple {9xy^{2}} + 2y^{3})}}$

• $:\implies{\mathrm { \: 4x^{3} + \purple {(7-4)x^{2}y} - \purple {(7+2)xy^{2}} + 2y^{3})}}$

• $:\implies{\mathrm { \:( 4x^{3} + \purple {7x^{2}y-4x^{2}y} - \purple {7xy^{2}+2xy^{2}} + 2y^{3}}}$

• $\sf{ Now\:Finding \:Common-\:Factors \:from\:each\:term-:}$

• $:\implies{\mathrm { \: 4x^{3} + 7x^{2}y-4x^{2}y - 7xy^{2}+2xy^{2} + 2y^{3}}}$

• $:\implies{\mathrm { \: \purple {4x^{2} + 7x^{2}y} \red{-4x^{2}y - 7xy^{2}}\pink{+2xy^{2} + 2y^{3}}}}$

• $:\implies{\mathrm { \: \purple {x^{2}( 4x + 7y)} \red{-xy(4x + 7y)}\pink{+2y^{2} ( x-y)}}}$

• $:\implies{\mathrm { \: \purple {(x^{2}-xy)} \red{(4x + 7y)}\pink{-2y^{2} ( x-y)}}}$

• $:\implies{\mathrm { \: \purple {(x^{2}-xy)} (4x + 7y)-2y^{2} \purple{( x-y)}}}$

• $:\implies{\mathrm { \: \purple {(x^{2}-xy)} (4x + 7y)-2y^{2} ( x-y)}}$

• $:\implies{\mathrm { \: \purple {x(x-y)} (4x + 7y)-2y^{2} ( x-y)}}$

• $:\implies{\mathrm { \: x\purple{(x-y)} (4x + 7y)-2y^{2} \purple{( x-y)}}}$

• $:\implies{\mathrm { \: \purple {(x-y)} \left\{x(4x + 7y)-2y^{2} \right\}}}$

• $:\implies{\mathrm { \: (x-y) \purple {\left\{x(4x + 7y)-2y^{2} \right\}}}}$

• $:\implies{\mathrm { \: (x-y) \purple {\left\{4x^{2} + 7xy-2y^{2} \right\}}}}$

• $:\implies{\mathrm { \: (x-y) \left\{4x^{2} + 7xy-2y^{2} \right\}}}$

• $\sf{\dag{ By\:Using \:Sum-\:Product \:pattern\:in\:new\:term-:}}$

• $:\implies{\mathrm { \: (x-y) \left\{4x^{2} \purple{+ 7xy}-2y^{2} \right\}}}$

• $:\implies{\mathrm { \: (x-y) \left\{4x^{2} \purple{+ (8-1)xy}-2y^{2} \right\}}}$

• $:\implies{\mathrm { \: (x-y) \left\{4x^{2} \purple{+ 8xy-xy}-2y^{2} \right\}}}$

• $\sf{ Now\:Finding \:Common-\:Factors \:from\:each\:term-:}$

• $:\implies{\mathrm { \: (x-y) \left\{ \blue{4x^{2} + 8xy}\purple{-xy-2y^{2}} \right\}}}$

• $:\implies{\mathrm { \: (x-y) \left\{\blue{4x(x+2y)}\purple{-y(x+2y)} \right\}}}$

• $\sf{ Now,\:Rewrite \:the\:factored\:term\:\:-:}$

• $:\implies{\mathrm { \: \red{ (x-y)}\blue{(4x-y)}\purple{(x+2y)} }}$

Therefore,

• $\underline {\boxed{\mathrm { \: Factors \:are\:-:(x-y)(4x-y)(x+2y)}}}$

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• Factorising second term -:

• $\mathrm {Second \:Term\:-: \:( x^{2} + 2xy - 2y^{2} )}$

• $:\implies{\mathrm { \:( x^{2} + 2xy - 2y^{2})}}$

• $\sf{\dag{ By\:Using \:Sum-\:Product \:pattern-:}}$

• $:\implies{\mathrm { \: x^{2} + \purple {2xy} - 2y^{2}}}$

• $:\implies{\mathrm { \: x^{2} + \purple {(2-1)xy} - 2y^{2}}}$

• $:\implies{\mathrm { \: x^{2} + \purple {2xy-xy} - 2y^{2}}}$

• $\sf{ Now\:Finding \:Common-\:Factors \:from\:each\:term-:}$

• $:\implies{\mathrm { \:\purple {x^{2} + 2xy} \red{-xy - 2y^{2}}}}$

• $:\implies{\mathrm { \:\purple {x(x + 2y)} \red{-y(x + 2y)}}}$

• $\sf{ Now,\:Rewrite \:the\:factored\:term\:\:-:}$

• $:\implies{\mathrm { \:\purple {(x + 2y)} \red{(x - y)}}}$

Therefore,

• $\underline {\boxed{\mathrm { \: Factors \:are\:-:(x-y)(x+2y)}}}$

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• H.C.F is also known as Common Factor

Therefore,

• $:\implies{\mathrm { \: Factors \:of\:1st\:term \:-: \purple{(x-y)}(4x-y)\purple{(x+2y)} }}$

• $:\implies{\mathrm { \: Factors \:of\:2nd\:term \:-: \purple{(x-y)(x+2y)} }}$

Hence ,

• $\underline {\boxed{\mathrm { The\:common\:\:factord\:in\:both\:terms \:are\:-:\bf{\red{(x-y)(x+2y)}}}}}$

• $\underline {\boxed{\mathrm { The\:Option \bf{\red{1}}\:or\:\bf{\red{(x-y)(x+2y)}}\:is\:correct}}}$

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