00:02
In this question, we are asked to write down the form or the partial fraction decomposition for the given rational functions.
00:11
This means that we just need to, you don't have to find the coefficients in the partial fraction decomposition.
00:18
And the first step would be to rewrite the denominator.
00:21
So first of all, note that the highest power of x in the numerator is less than the highest power of x in the denominator.
00:30
In the denominator, if you multiply x squared and x to the fourth, you're going to get x to the 6 power.
00:37
And 6 is greater than 5, right? so this means that this is a proper rational function.
00:49
And in this case, first we need to factor the denominator as much as possible.
00:58
And the first step would be to factor out x from the first pair of parentheses.
01:06
Going to get x multiplied by x minus 1 and in the second pair of parentheses we can errat it as x squared plus 2 squared all right now the first two factors are linear factors this and this are a linear and the expression inside the square in the second factor is an irreducible quadratic reducible quadratic and this is irreducible because if you try to solve x squared plus 2 equals 0, it has no solutions.
02:02
X squared plus 2 equals 0 implies that x squared plus 2 equals 0 implies that x squared equals to negative 2, which is impossible.
02:25
This is impossible.
02:27
And therefore, x squared plus 2 is an irreducible quadrate.
02:31
So this means that our partial fraction decomposition will look like a over x plus b over x minus 1, one fraction for each linear term, and for the reducible quadratic, we are going to have two fractions because it's squared, because of the square here...