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State whether the given statement is true/false.

The polynomial $x 2 + 2 x + 2$ is a factor of $x 4 + 3 x 3 + 7 x 2 + x + 13$ .

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True

False

answer is B.

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Detailed Solution

Given polynomials are

x 2 + 2 x + 2

and

x 4 + 3 x 3 + 7 x 2 + x + 13

. We need to find out whether

x 2 + 2 x + 2

is a factor of

x 4 + 3 x 3 + 7 x 2 + x + 13

or not.
When a polynomial is divided by a polynomial and leaves the remainder 0, then it is the factor of that polynomial.
We will use the long division method for performing the division.
Question Image

Remainder is

− 7 x + 7

.
As remainder is not

0

, so

x 2 + 2 x + 2

is not a factor of the polynomial

3 x 4 + 5 x 3 − 7 x 2 + 2 x + 2

.
Hence, the statement that the polynomial

x 2 + 2 x + 2

is a factor of

x 4 + 3 x 3 + 7 x 2 + x + 13

is incorrect.
Therefore, the correct statement would be that the polynomial

x 2 + 2 x + 2

is not the factor of the polynomial

x 4 + 3 x 3 + 7 x 2 + x + 13

.
Therefore, option 2 is correct.

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