If the polynomial P(x) = x^4 + ax^3 - 7x^2 + bx + 12 is completely divisible by x^2 - 2x -… — JAMB UTME

If the polynomial $P(x) = x^4 + ax^3 - 7x^2 + bx + 12$ is completely divisible by $x^2 - 2x - 3$ , what is the value of $a - b$ ?

$-10$ Cevap
B
$10$
C
$-5$
D
$5$

Cevap

The value of

a - b

-10

Factoring

x^2 - 2x - 3

gives

(x - 3)(x + 1)

. By the Factor Theorem,

P(3) = 0

and

P(-1) = 0

. Substituting these into

P(x)

produces the linear system

9a + b = -10

and

a + b = 6

. Solving this system yields

a = -2

and

b = 8

. Subtracting gives

a - b = -2 - 8 = -10

Adım Adım Çözüm

Factor the quadratic divisor to find the roots.

x^2 - 2x - 3 = (x - 3)(x + 1)

, so the roots are

x = 3

and

x = -1

By the Factor Theorem, if a polynomial is divisible by a quadratic expression,

P(x)

must evaluate to zero at each root of the divisor.

Set up equations by evaluating

P(3) = 0

and

P(-1) = 0

For

x = 3

3^4 + a(3)^3 - 7(3)^2 + b(3) + 12 = 0 \implies 81 + 27a - 63 + 3b + 12 = 0 \implies 9a + b = -10

.
For

x = -1

(-1)^4 + a(-1)^3 - 7(-1)^2 + b(-1) + 12 = 0 \implies 1 - a - 7 - b + 12 = 0 \implies a + b = 6

Evaluating the polynomial at each root yields a system of two linear equations in variables

a

and

b

Solve the simultaneous equations for

a

and

b

Subtracting

(a + b = 6)

from

(9a + b = -10)

gives

8a = -16 \implies a = -2

.
Substituting

a = -2

into

a + b = 6

gives

-2 + b = 6 \implies b = 8

Elimination isolates

a

, allowing both

a

and

b

to be uniquely determined.

Calculate

a - b

a - b = -2 - 8 = -10

This computes the required expression value.

Anahtar Kavram

Factor Theorem for quadratic divisors

Tahmini Süre:2m 0s

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