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Question

If x+y and y+3x are two factor of the expression λx3μx2y+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 y3x
Let f(x)=λx3μx2y+xy2+y3
As x+y is factor
f(y)=λ(y)3μ(y)2y+(y)y2+y3y3λμy2yy3+y3=0
λ+μ=0 ...(1)
And as y+3x is factor
f(y3)=λ(y3)3μ(y3)2y+(y3)y2+y3λy327μy39y33+y3=0λy33μy39y3+27y3=0
λ+3μ=18 ...(2)
Solving (1) and (2), we get
λ=9,μ=9
Then
f(x)=9x39x2y+xy2+y3=(x+y)(y+3x)(y3x)
Hence, option 'B' is correct.

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