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The partial fraction decomposition has the form:

\[
\frac{5x^{4} - 34x^{3} + 70x^{2} - 33x - 19}{(x-3)^{2}} = f(x) + \frac{g(x)}{x-3} + \frac{h(x)}{(x-3)^{2}}
\]

Answer :

The expression cannot be further simplified using partial fraction decomposition. The given expression can be rewritten as the sum of three fractions using partial fraction decomposition.

We want to find the values of f(x), g(x), and h(x) that satisfy the equation:

[tex](5x^4 - 34x^3 + 70x^2 - 33x - 19) / (x - 3)^2 = f(x) + g(x) / (x - 3) + h(x) / (x - 3)^2[/tex]
To determine the values of f(x), g(x), and h(x), we need to clear the denominators. We can do this by multiplying both sides of the equation by [tex](x - 3)^2[/tex]:

[tex](5x^4 - 34x^3 + 70x^2 - 33x - 19) = (x - 3)^2 * f(x) + (x - 3) * g(x) + h(x)[/tex]
Expanding the right side of the equation:

[tex]5x^4 - 34x^3 + 70x^2 - 33x - 19 = (x^2 - 6x + 9) * f(x) + (x - 3) * g(x) + h(x)[/tex]
Now, let's compare the coefficients of like terms on both sides of the equation.

For the [tex]x^4[/tex] term, the coefficient on the left side is 5, while there are no [tex]x^4[/tex] terms on the right side. This implies that f(x) = 0.

For the [tex]x^3[/tex] term, the coefficient on the left side is -34, while there are no [tex]x^3[/tex] terms on the right side. This implies that g(x) = 0.

For the [tex]x^2[/tex] term, the coefficient on the left side is 70, and on the right side, we have ([tex]x^2[/tex] - 6x + 9) * f(x). Since f(x) is 0, the coefficient of the [tex]x^2[/tex] term must also be 0. This gives us the equation:

[tex]70 = 0 * (x^2 - 6x + 9)[/tex]

Simplifying, we find that 70 = 0, which is not possible.

Therefore, there is no solution for h(x).

In summary, the partial fraction decomposition of the given expression is:

[tex](5x^4 - 34x^3 + 70x^2 - 33x - 19) / (x - 3)^2 = 0 + 0 / (x - 3) + 0 / (x - 3)^2[/tex]


Since there is no solution for f(x), g(x), and h(x), the expression cannot be further simplified using partial fraction decomposition.

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