If x + a is a common factor of f(x) = x2 + x – 6 and g(x) = x2 + 3x

1.	If x + a is a common factor of f(x) = x2 + x – 6 and g(x) = x2 + 3x – 18, then find the value of a.
Answer» Given polynomials are f(x) = x² + x – 6 and g(x) = x² + 3x – 18 f(x) and g(x) having common factors (x + a), So x + a = 0 ⇒ x = -a f(-a) = (-a)² + (-a) – 6 = a² – a – 6 g(-a) = (-a)² + 3(-a) – 18 = a² – 3a – 18 f(-a) = g(-a) a² – a – 6 = a² – 3a – 18 a² – a² – a + 3a = -18 + 6 2a = -12 a = -6.

If x + a is a common factor of f(x) = x2 + x – 6 and g(x) = x2 + 3x – 18, then find the value of a.

Answer»

Given polynomials are f(x) = x² + x – 6

and g(x) = x² + 3x – 18

f(x) and g(x) having common factors

(x + a), So

x + a = 0 ⇒ x = -a

f(-a) = (-a)² + (-a) – 6

= a² – a – 6

g(-a) = (-a)² + 3(-a) – 18

= a² – 3a – 18

f(-a) = g(-a)

a² – a – 6 = a² – 3a – 18

a² – a² – a + 3a = -18 + 6

2a = -12

a = -6.