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Question

# If x+y and y+3x are two factor of the expression λx3−μx2y+xy2+y3, then the third factor is

A
y+3x
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B
y3x
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C
yx
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D
none of these
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Solution

## The correct option is B y−3xLet f(x)=λx3−μx2y+xy2+y3As x+y is factor⇒f(−y)=λ(−y)3−μ(−y)2y+(−y)y2+y3⇒−y3λ−μy2y−y3+y3=0⇒λ+μ=0 ...(1)And as y+3x is factor⇒f(−y3)=λ(−y3)3−μ(−y3)2y+(−y3)y2+y3⇒−λy327−μy39−y33+y3=0⇒−λy3−3μy3−9y3+27y3=0⇒λ+3μ=18 ...(2)Solving (1) and (2), we getλ=−9,μ=9Thenf(x)=−9x3−9x2y+xy2+y3=(x+y)(y+3x)(y−3x)Hence, option 'B' is correct.

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