Question

if x²+x-12 divides p(x)= x³+ax²+bx-84 exactly ,find the a and b.

Answer 1

x^2 + x + 12 divides p ( x ) exactly means x^2 + x + 12 is a factor of p ( x ) factors of x^2 + x + 12 are also factors of p ( x )

x^2 + x - 12

x^2 + 4x - 3 x -12

= x ( x+ 4 ) - 3 ( x + 4 )

= ( x + 4 ) ( x - 3 )

so using factors theorem

x = 3

x^3 + ax^2 + bx - 84

( 3 )^3 + a( 3 )^2 + b(3) - 84 = 0

27 + 9a + 3b - 84 = 0

9a + 3b = 54

3a + b =18. .... ( l )

when

x = - 4

x^3 + ax^2 + bx - 84 = 0

( - 4 )^3 + a ( - 4 )^2 + ( - 4 )b - 84 =0

- 64 + 16a - 4 b - 84 = 0

16 a - 4b = 148

4a - b = 37. ... ( ll )

adding the equations ( l ) & ( ll ) we get

9a = 55

a = 55/9