What is the remainder when the polynomial P(x) = x^3 - 4x^2 + 5x - 2 is divided by (x + 1)… — JAMB UTME

What is the remainder when the polynomial $P(x) = x^3 - 4x^2 + 5x - 2$ is divided by $(x + 1)$ ?

-12

According to the Remainder Theorem, dividing a polynomial

P(x)

by a linear divisor

(x - a)

yields a remainder equal to

P(a)

. Here, the divisor is

(x + 1)

, which corresponds to

a = -1

. Substituting

x = -1

into

P(x) = x^3 - 4x^2 + 5x - 2

gives

(-1)^3 - 4(-1)^2 + 5(-1) - 2 = -1 - 4 - 5 - 2 = -12

. Thus, the value

-12

is correct.

Apply the Remainder Theorem

To find the remainder when

P(x)

is divided by

(x + 1)

, evaluate

P(-1)

by setting

x + 1 = 0 \implies x = -1

By the Remainder Theorem, dividing

P(x)

(x - a)

leaves a remainder equal to

P(a)

Substitute

x = -1

into the polynomial

P(x) = x^3 - 4x^2 + 5x - 2

P(-1) = (-1)^3 - 4(-1)^2 + 5(-1) - 2

Replace each instance of

x

with

-1

Simplify the powers and terms

P(-1) = -1 - 4(1) - 5 - 2 = -1 - 4 - 5 - 2 = -12

Compute arithmetic operations following standard order of operations.

Remainder Theorem

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