Question

If (x-2) & (x+3) are both factors of the polynomial (x)=x^{​​​​​​3}+ax^{​​​​​2}+bx-12 find the values of a and b.

Answer 1

Consider, x^{​​​​​​3}+ax^{​​​​​2}+bx-12 = 0

x-2 is a factor. So, x=2 is a solution. Put x=2
2^3 + a (2^2) + b (2) - 12 = 0
4a + 2b = 4
2a + b = 2 ......(I)

x+3 is a factor. So, x= -3 is a solution. Put x= -3
-3^3 + a( -3^2) + b (-3) - 12 = 0
9a -3b = 39
3a - b = 13 ......(II)

(I) + (II) => 5a = 15 => a = 3
From (I), 2 (3) + b = 2 => b = -4