Question

find the values of m which X square + 3 + X + MY - M has two linear factors in X and Y with integers coefficient

Answer 1

Let, x²+3xy+x+my-m=(ax+by+c)(ex+fy+g)

Since the given expression does not contain y² then one of the linear factors must not contain the term containing y. Therefore,

x²+3xy+x+my-m=(ax+by+c)(ex+g)

or, x²+3xy+x+my-m=aex²+bexy+cex+agx+bgy+cg

Equating the coefficients from both sides,

ae=1 ----------------------(1)

be=3 ----------------------(2)

ce+ag=1 -----------------(3)

bg=m ---------------------(4)

cg=-m --------------------(5)

Since all the coefficients are integers then from (1),

a=e=1

∴, from (2), b=3

Putting in(3),

c+g=1 ----------------(6)

Now dividing (4) by (5),

b/c=-1

or, c=-b

or, c=-3

∴, from (6),

-3+g=1

or, g=4

Putting in (5),

-m=-3×4

or, m=12 Ans.