Given that (x + 2) is a factor of the polynomial P(x) = 2x^3 - x^2 + ax + b, and that divi… — JAMB UTME

Given that $(x + 2)$ is a factor of the polynomial $P(x) = 2x^3 - x^2 + ax + b$ , and that dividing $P(x)$ by $(2x - 1)$ leaves a remainder of $15$ , what is the value of $a + b$ ?

A
$6$
B
$11$
$14$ Answer
D
$-1$

Answer

The value of

a + b

14

Applying the Factor Theorem with

x = -2

gives the equation

-2a + b = 20

. Applying the Remainder Theorem with

x = \frac{1}{2}

gives the equation

a + 2b = 30

. Solving these simultaneous equations yields

a = -2

and

b = 16

. Adding

a

and

b

gives

14

Step-by-Step Solution

Apply the Factor Theorem for

(x + 2)

-2a + b = 20

Since

(x + 2)

is a factor of

P(x)

P(-2) = 0

. Substituting

x = -2

yields

2(-2)^3 - (-2)^2 + a(-2) + b = 0

, which simplifies to

-16 - 4 - 2a + b = 0

-2a + b = 20

Apply the Remainder Theorem for

(2x - 1)

a + 2b = 30

Dividing

P(x)

(2x - 1)

leaves a remainder of

15

, so

P\left(\frac{1}{2}\right) = 15

. Substituting

x = \frac{1}{2}

yields

2\left(\frac{1}{8}\right) - \frac{1}{4} + \frac{a}{2} + b = 15

, which simplifies to

\frac{a}{2} + b = 15

a + 2b = 30

Solve the system of linear equations for

a

and

b

a = -2

and

b = 16

From step 1,

b = 2a + 20

. Substituting this into step 2 gives

a + 2(2a + 20) = 30 \implies 5a + 40 = 30 \implies 5a = -10 \implies a = -2

. Substituting

a = -2

into

b = 2a + 20

gives

b = 16

Calculate

a + b

14

Summing the calculated values gives

a + b = -2 + 16 = 14

Key Concept

Factor Theorem and Remainder Theorem for Polynomials

Question

Answer

Step-by-Step Solution

Key Concept

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