Question

Given that, 2x^3 + 5x^2+7x + 23 = A(x+1)^3+ B(x + 1)^2 +Cx+ 22 for all values of x, evaluate A, B and C. Hence, state the remainder when 2x^3 +5x^2 + 7x+23 is divided by x^2 + 2x +1.​

Answer 1

A = 2

B = -1

C = 3

remainder= 19

Explanation:

A(x+1)³ + B(x+1)² + Cx + 22 =

A(x³ + 3x² + 3x + 1) + B(x² + 2x + 1) + Cx + 22 =

Ax³ + (3A + B)x² + (3A + 2B + C)x + (A + B + 22) =

2x³ + 5x² + 7x + 23

→

A = 2

3A + B = 5 → 3(2) + B = 5 → B = 5 - 6 → B = -1

3A + 2B + C = 7 → 3(2) + 2(-1) + C = 7 → C = 7 - 6 + 2 →

C = 3

remainder :

(2x³ + 5x² + 7x + 23 )/ (x² + 2x + 1) =

(2x³ + 5x² + 7x + 23 )/ (x + 1)²

→

(x + 1)² = 0 → x = -1

2(-1)³ + 5(-1)² + 7(-1) + 23 = -2 + 5 - 7 + 23 = 19