Question

Using factor theorem,show that x^2+5x+6 is a factor of x^4+5x^2+9x+15x+18

Answer 1

Answer:

We have,

x2 + 5x + 6

= x2 + 3x + 2x + 6

= x (x + 3) +2 (x + 3)

= (x + 2) (x + 3)

Let p(x) = x4 + 5x3 + 9x2 + 15x + 18

Put x = -2

p(-2) = (-2)4 + 5(-2)3 + 9(-2)2 + 15 (-2) + 18

= 16 - 40 + 36 - 30 + 18

= 70 - 70 = 0  (x + 2) is a factor

Put x = -3

p(-3) = (-3)4 + 5(-3)3 + 9(-3)2 + 15 (-3) + 18

= 81 - 135 + 81 - 45 + 18

= 180 - 180 = 0  (x + 3) is a factor

Hence x2 + 5x + 6

Is a factor of p(x).