Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm:
$g(x)\ =\ x^3\ –\ 3x\ +\ 1;\ f(x)\ =\ x^5\ –\ 4x^3\ +\ x^2\ +\ 3x\ +\ 1$


Given:


$g(x)\ =\ x^3\ –\ 3x\ +\ 1$ and $f(x)\ =\ x^5\ –\ 4x^3\ +\ x^2\ +\ 3x\ +\ 1$.


To do:


We have to check whether $g(x)$ is a factor of $f(x)$ by applying the division algorithm.

 

Solution:


On applying the division algorithm, 


Dividend$f(x)\ =\ x^5\ –\ 4x^3\ +\ x^2\ +\ 3x\ +\ 1$

 

Divisor$g(x)\ =\ x^3\ –\ 3x\ +\ 1$


If $g(x)$ is a factor of $f(x)$ then the remainder on long division should be $0$.

 

$x^3-3x+1$)$x^5-4x^3+x^2+3x+1$($x^2-1$

                       $x^5-3x^3+x^2$

               -------------------------------

                              $-x^3+3x+1$

$-x^3+3x-1$

-------------------

$0$


Therefore, $g(x)$ is a factor of $f(x)$.

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Updated on: 10-Oct-2022

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