Find the values of ‘m’ for which x2+3xy+x+my-m has two linear factors in x and y, with integer coefficients.

# Find the values of ‘m’ for which ${x}^{2}+3\mathit{xy}+x+\mathit{my}-m$ has two linear factors in x and y, with integer coefficients.

1. A
m = 1 or m = 11
2. B
m = 0 or m = 12
3. C
m = 10 or m = 0
4. D
None

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### Solution:

Given equation is,
${x}^{2}+3\mathit{xy}+x+\mathit{my}-m$
Suppose
Equating the coefficients:
ae=1            ….(1)
be=3              ….(2)
ce+ag=1         ...…(3)
bg=m              ….(4)
cg=−m            ….(5)
The value of a=e=1, and b=3 to satisfy first two equation. Thus, from equation 4,
3q=m
After putting this value in equation 3, we get,
e+q=1          ….(6)
Now, $\frac{b}{c}=-1$ or
c=−b, c=−3
Now, from equation (6), −3+g=1  g=4
Putting in (5):
−m=−3×4
m=12 Option 2 is correct.

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