Find the values of ‘m’ for which x2 + 3xy + x + my – m has two

1.	Find the values of ‘m’ for which x2 + 3xy + x + my – m has two linear factors in x and y, with integer coefficients.
Answer» Given equation is x² + 3xy + x + my – m ……….(1) Let the two linear equations in x and y be (x + 3y + a) and (x + 0y + b). Then (x + 3y + a) (x + 0y + b) = x² + 0xy + bx + 3xy + 0y² + 3by + ax + 0y + ab = x² + bx + ax + 3xy + 3by + ab ………….. (2) Comparing equation (2) with (1), x² + 3xy + x + my – m = x² + (a + b)x + 3xy + 3by + ab Equating the like terms on both sides, ab = – m ………….. (3) (a + b)x = x ⇒ a + b = 1…………. (4) 3by = my ⇒ 3b = m ⇒ b = m/3 Substitute ‘b’ value in equation (4), a = 1−m/3 = (3 − m)/3 ab = -m [ ∵ from (3)] put a & b value then , ((3−m)/3)(m/3) = -m (3m − m²)/9 = -m ⇒ 3m – m² = – 9m ⇒ m² – 12m = 0 ⇒ m(m – 12) = 0 ⇒ m = 0 (or) m = 12 lf m = 12 ∴ b = 12/3 = 4&a = (3−m)/3 = (3−12)/3 = −9/3 = -3 ∴ Linear factors are (x + 3y – 3), (x + 4) If m = 0 b = 0/3 = 0 & a = (3−0)/3 = 3/3 = 1 ∴ Linear factors are (x + 3y + 1), x.

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

Answer»

Given equation is x² + 3xy + x + my – m ……….(1)

Let the two linear equations in x and y be (x + 3y + a) and (x + 0y + b).

Then (x + 3y + a) (x + 0y + b)

= x² + 0xy + bx + 3xy + 0y² + 3by + ax + 0y + ab

= x² + bx + ax + 3xy + 3by + ab ………….. (2)

Comparing equation (2) with (1),

x² + 3xy + x + my – m = x² + (a + b)x + 3xy + 3by + ab

Equating the like terms on both sides,

ab = – m ………….. (3)

(a + b)x = x

⇒ a + b = 1…………. (4)

3by = my ⇒ 3b = m ⇒ b = m/3

Substitute ‘b’ value in equation (4),

a = 1−m/3 = (3 − m)/3

ab = -m

[ ∵ from (3)]

put a & b value then ,

((3−m)/3)(m/3) = -m

(3m − m²)/9 = -m

⇒ 3m – m² = – 9m

⇒ m² – 12m = 0

⇒ m(m – 12) = 0

⇒ m = 0 (or) m = 12

lf m = 12

∴ b = 12/3 = 4&a = (3−m)/3 = (3−12)/3

= −9/3 = -3

∴ Linear factors are (x + 3y – 3), (x + 4) If m = 0

b = 0/3 = 0 & a = (3−0)/3 = 3/3 = 1

∴ Linear factors are (x + 3y + 1), x.