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Find the GCD (greatest common divisor) of the polynomials $x^{4} + 3 x^{3} - x - 3$ and $x^{3} + x^{2} - 5 x + 3 .$

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x^{2} + 2 x - 3

x^{2} + 5 x - 3

x^{2} + 2 x + 3

x^{2} + 2 x - 5

answer is A.

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Detailed Solution

greatest common divisor
Given polynomials are,

x^{4} + 3 x^{3} - x - 3

and

x^{3} + x^{2} - 5 x + 3

Suppose,

f (x) = x^{4} + 3 x^{3} - x - 3

Here, highest power of variable x is 4. Thus, degree of this function is 4.

g (x) = x^{3} + x^{2} - 5 x + 3

Here, highest power of variable x is 3. Thus, degree of this function is 3.
So,
Degree of

f (x) >

degree of

g (x) .

Thus, in our case divisor is

x^{3} + x^{2} - 5 x + 3

while dividend is

x^{4} + 3 x^{3} - x - 3 .

The reminder is,

3 x^{2} + 6 x - 9

Divide it by 3,

x^{2} + 2 x - 3

This is the greatest common divisor.
Option 1 is correct.

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