Example 2:

Factorize the quartic polynomial $x^4\ -\ 10x^3\ +\ 21x^2\ +\ 40x\ -\ 100$ using rational root test

Solution:

Using rational root test we find out the possible roots of the quartic polynomial. The value of $p$ is $100$ and that of $q$ is $1$.

So, $\frac{p}{q}$ = $\pm$ $\frac{100}{1}$ = $\pm 100$. The factors are $\pm\ (1, 2, 5, 10, 20, 25, 50, 100)$. Let us choose $5$ and test whether its a root or not.

$P(5)$ = $5^4\ -\ 10\ \times\ 5^3\ +\ 21\ \times\ 5^2\ +\ 40\ \times\ 5\ -\ 100$ = $0$

Therefore, $x - 5$ is a factor of the quartic polynomial. Using synthetic division method we try to find out the further roots.

The polynomial equation we get is $x^3\ -\ 5x^2\ -\ 4x\ +\ 20$. Factorizing it further we get:

$x^3\ -\ 5x^2\ -\ 4x\ +\ 20$

$x^2\ (x - 5)\ -\ 4\ (x - 5)$

$(x^2 - 4)\ (x - 5)$

$(x + 2)\ (x - 2)\ (x - 5)$

Thus, the roots of the quartic polynomial $x^4\ -\ 10x^3\ +\ 21x^2\ +\ 40x\ -\ 100$ are $-5,\ -2,\ 2$ and $5$.