Find the values of b and c given that (x-2)(x^2+bx+c)=x^3+2x^2-5x-6 for all values of x

Question

Find the values of b and c given that
`(x-2)(x^2+bx+c)=x^3+2x^2-5x-6` for all values of x

Answer 1

b=4 c=3

Explanation:

Equation:

(x-2)(x^2+bx+c) = x^3+2x^2=5x-6

Distribute left side:

(x-2)(x^2+bx+c)= x^3+bx^2+cx -2x^2 -2bx -2c

Group:

x^3+ (b-2)x^2 + (c-2b)x -2c

Solve the equation:

b-2=2

c-2b=-5

-2c=-6

Answer:

b=4

c=3