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The linear polynomial $x + 1$ is a factor of which of the following polynomials?

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x 3 − 2 x 2 − x − 2

x 3 − 2 x 2 − x + 2

x 3 − 2 x 2 + x − 2

x 3 − 2 x 2 + x + 2

answer is D.

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Detailed Solution

It is given that the linear polynomial

x + 1

is a factor of a polynomial.
We know by Factor theorem that if

x − k

is a factor of the given polynomial

p (x)

, then

p (k) = 0

and if it is not a factor then

p k = Remainder .

Therefore, we will check for the polynomial that gives 0 at

x = − 1.

Option (1):
Checking for

x 3 − 2 x 2 + x + 2.

It gives,

(− 1) 3 − 2 (− 1) 2 + (− 1) + 2 = − 1 − 2 − 1 + 2 = − 2

Therefore,

x + 1

is not its factor.
Option (2):
Checking for

x 3 − 2 x 2 + x − 2.

It gives,

(− 1) 3 − 2 (− 1) 2 + (− 1) − 2 = − 1 − 2 − 1 − 2 = − 6

Therefore,

x + 1

is not its factor.
Option (3):
Checking for

x 3 − 2 x 2 − x − 2.

It gives,

(− 1) 3 − 2 (− 1) 2 − (− 1) − 2 = − 1 − 2 + 1 − 2 = − 4

Therefore,

x + 1

is not its factor.
Option (4):
Checking for

x 3 − 2 x 2 − x + 2.

It gives,

(− 1) 3 − 2 (− 1) 2 − (− 1) + 2 = − 1 − 2 + 1 + 2 = 0

Therefore,

x + 1

is a factor of the polynomial

x 3 − 2 x 2 − x + 2.

Thus, option (4) is correct.

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The linear polynomial x+1 is a factor of which of the following polynomials?

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The linear polynomial $x + 1$ is a factor of which of the following polynomials?