Find the values of ‘m’ for which x2+3xy+x+my-m has two linear factors in x and y, with integer coefficients. - Sri Chaitanya Infinity Learn Best Online Courses for NCERT Solutions, CBSE, ICSE, JEE, NEET, Olympiad and Class 6 to 12

A
m = 1 or m = 11
B
m = 0 or m = 12
C
m = 10 or m = 0
D
None

Solution:

Given equation is,

x^{2} + 3 xy + x + my - m

Suppose

x^{2} + 3 xy + x + my - m = (ax + by + c) (ex + fy + g)

x^{2} + 3 xy + x + my - m = ae x^{2} + bexy + cex + agx + bgy + cg

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.

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

Solution:

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