Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:
$x^3 -3x + 1, x^5 - 4x^3 + x^2 + 3x + l$
Given:
$x^3\ –\ 3x\ +\ 1$ and $ x^5\ –\ 4x^3\ +\ x^2\ +\ 3x\ +\ 1$.
To do:
We have to check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial.
Solution:
On applying the division algorithm,
Let dividend$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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