What is the remainder when the polynomial P(x) = 2x^3 - 5x^2 + 7x - 3 is divided by (2x - … — JAMB UTME

What is the remainder when the polynomial $P(x) = 2x^3 - 5x^2 + 7x - 3$ is divided by $(2x - 1)$ ?

The remainder is

-\frac{1}{2}

By the Remainder Theorem, when

P(x)

is divided by

(ax - b)

, the remainder is

P\left(\frac{b}{a}\right)

. Setting

2x - 1 = 0

yields

x = \frac{1}{2}

. Evaluating

P\left(\frac{1}{2}\right) = 2\left(\frac{1}{8}\right) - 5\left(\frac{1}{4}\right) + 7\left(\frac{1}{2}\right) - 3 = \frac{1}{4} - \frac{5}{4} + \frac{7}{2} - 3 = -\frac{1}{2}

Find the root of the linear divisor

2x - 1 = 0 \implies x = \frac{1}{2}

According to the Remainder Theorem, dividing

P(x)

by a linear divisor

(ax - b)

yields a remainder of

P\left(\frac{b}{a}\right)

Substitute

x = \frac{1}{2}

into

P(x) = 2x^3 - 5x^2 + 7x - 3

P\left(\frac{1}{2}\right) = 2\left(\frac{1}{2}\right)^3 - 5\left(\frac{1}{2}\right)^2 + 7\left(\frac{1}{2}\right) - 3

Evaluating the polynomial at the root of the divisor determines the remainder.

Calculate the arithmetic value

P\left(\frac{1}{2}\right) = 2\left(\frac{1}{8}\right) - 5\left(\frac{1}{4}\right) + \frac{7}{2} - 3 = \frac{1}{4} - \frac{5}{4} + \frac{7}{2} - 3 = -1 + \frac{7}{2} - 3 = -4 + 3.5 = -0.5 = -\frac{1}{2}

Simplifying the fractional terms gives the final remainder value.

Remainder Theorem for Linear Divisors

(ax - b)

Question