Question

What number must be added to x³ + 5x² − 6x so that the resulting polynomial is exactly divisible by (x – 2)?

Answer 1

Let the number added be k

f(x) = x³ + 5x² − 6x + k

Let x – 2 = 0

x = 2

f(2) = 2³ + 5(2)² − 6(2) + k

= 8 + 5(4) − 12 + k

= 8 + 20 − 12 + k

= 16 + k

Set the remainder f(2) equal to zero to find the value of k

f(2) = 0

16 + k = 0

k = −16

∴ The number that must be added to the polynomial so that the resulting polynomial is exactly divisible by (x – 2) is −16.