Given that (x - 2) is a factor of the polynomial P(x) = x^3 + kx^2 - 5x + 6, find the rema… — JAMB UTME

Given that $(x - 2)$ is a factor of the polynomial $P(x) = x^3 + kx^2 - 5x + 6$ , find the remainder when $P(x)$ is divided by $(x + 3)$ .

Cevap: -15

The remainder when

P(x)

is divided by

(x + 3)

-15

According to the Factor Theorem, since

(x - 2)

is a factor of

P(x) = x^3 + kx^2 - 5x + 6

, setting

x = 2

yields

P(2) = 0

. This gives

8 + 4k - 10 + 6 = 0

, which simplifies to

4k + 4 = 0

, giving

k = -1

. The polynomial is therefore

P(x) = x^3 - x^2 - 5x + 6

. By the Remainder Theorem, dividing

P(x)

(x + 3)

produces a remainder of

P(-3)

. Evaluating

P(-3) = (-3)^3 - (-3)^2 - 5(-3) + 6 = -27 - 9 + 15 + 6 = -15

Apply the Factor Theorem to determine the unknown constant

k

k = -1

(x - 2)

is a factor of

P(x)

, then

P(2) = 0

. Substituting

x = 2

gives

2^3 + k(2)^2 - 5(2) + 6 = 0 \implies 4k + 4 = 0 \implies k = -1

Substitute

k = -1

into the original polynomial to get the full expression.

P(x) = x^3 - x^2 - 5x + 6

Replacing

k

with

-1

defines

P(x)

completely.

Apply the Remainder Theorem to find the remainder when

P(x)

is divided by

(x + 3)

Remainder is

-15

By the Remainder Theorem, dividing

P(x)

(x + 3)

leaves a remainder equal to

P(-3)

. Calculating

P(-3) = (-3)^3 - (-3)^2 - 5(-3) + 6 = -27 - 9 + 15 + 6 = -15

Factor and Remainder Theorems for Polynomials

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