When the polynomial P(x) = x^3 - 2x^2 + ax + 8 is divided by (x - 3), the remainder is 14.… — JAMB UTME

When the polynomial $P(x) = x^3 - 2x^2 + ax + 8$ is divided by $(x - 3)$ , the remainder is $14$ . What is the value of the constant $a$ ?

Answer: -1

The value of the constant

a

-1

According to the Remainder Theorem, dividing

P(x)

(x - 3)

means the remainder is

P(3)

. Evaluating

P(3) = 3^3 - 2(3)^2 + 3a + 8 = 27 - 18 + 3a + 8 = 17 + 3a

. Setting this equal to the remainder

14

gives

17 + 3a = 14

, which simplifies to

3a = -3

and yields

a = -1

Apply the Remainder Theorem

The remainder when

P(x)

is divided by

(x - 3)

is equal to

P(3)

The Remainder Theorem states that dividing a polynomial

P(x)

(x - c)

yields a remainder equal to

P(c)

Substitute

x = 3

into the polynomial and set equal to the given remainder

3^3 - 2(3)^2 + a(3) + 8 = 14

Setting the value of

P(3)

equal to

14

allows us to form a linear equation for the unknown constant

a

Simplify numerical terms in the equation

27 - 18 + 3a + 8 = 14 \implies 17 + 3a = 14

Evaluate exponents and multiplication to isolate the term containing

a

Solve for

a

3a = -3 \implies a = -1

Subtract

17

from both sides and divide by

3

Polynomial Remainder Theorem

Question