Question

What number must be subtracted from the polynomial 4x³ − 6x² + x + 17 so that when the resulting polynomial is divided by x − 3, it leaves a remainder 10?

Answer 1

Let the number subtracted be k

f(x) = 4x³ − 6x² + x + 17 − k

Let x − 3 = 0

x = 3

When f(x) is divided by x − 3, the remainder is,

f(3) = 10

⇒ 4(3)³ − 6(3)² + 3 + 17 − k = 10

⇒ 4(27) − 6(9) + 3 + 17 − k = 10

⇒ 108 − 54 + 3 + 17 − k = 10

⇒ 74 − k = 10

⇒ −k = 10 − 74

⇒ −k = −64

⇒ k = 64

∴ The number that must be subtracted is 64.