Partial Fraction Decomposition: A Step-by-Step Guide to Simplifying 235 / (x^4)
When dealing with rational expressions, particularly those involving polynomials in the denominator, partial fraction decomposition can be a powerful and useful technique. This article will guide you through the process of decomposing the complex fraction 235 / (x^4) into simpler fractions, making it easier to solve polynomial equations and integrals.
Understanding Partial Fraction Decomposition
Partial fraction decomposition is the process of breaking a rational function into the sum of simpler rational functions, each with a linear or irreducible quadratic factor in the denominator. The aim is to simplify the fraction into a form that can be easier to work with, such as in integration or differentiation. In our case, we will be dealing with the fraction 235 / (x^4).
Simplifying the Expression 235 / (x^4)
The given expression is 235 / (x^4). This can be simplified into a sum of partial fractions by first factoring the denominator, which, in this case, is already a simple power of x. We will decompose it into individual fractions with lower powers of x in the denominators.
Step 1: Factor the Denominator
The denominator of the given fraction is x^4, which can be factored as:
x^4 x * x * x * x
However, for simplicity and clarity, we will consider each factor as one: (x - 0).
Step 2: Write the Expression as a Sum of Partial Fractions
Since the denominator is a single power of x, we can write the given fraction as a sum of fractions with linear terms in the denominator. Generally, for a term of the form 1 / (x^n), you would write it as:
1 / (x^n) A_1 / (x - a_1) A_2 / (x - a_2) ... A_n / (x - a_n)
In our case, since we only have x^4, we can write:
235 / x^4 A_1 / x A_2 / (x^2) A_3 / (x^3) A_4 / (x^4)
Our goal is to find the coefficients A_1, A_2, A_3, and A_4.
Step 3: Determine the Coefficients
To determine the coefficients, we equate:
235 A_1 * x^3 A_2 * x^2 A_3 * x A_4
We can solve for the coefficients by plugging in different values for x or using algebraic methods. However, in simpler cases like this, it's often easier to directly substitute. Here, we can use the fact that the expression is a polynomial of degree 3, so we can solve it directly.
235 A_4
Hence, we can see that:
A_4 235
Now, we need to find the other coefficients by substituting values for x. Let's start by substituting x 1 to find A_1:
235 A_1 * 1^3 A_2 * 1^2 A_3 * 1 235
235 A_1 A_2 A_3 235
0 A_1 A_2 A_3
Now, substituting x 0 to find A_3:
235 A_1 * 0^3 A_2 * 0^2 A_3 * 0 235
235 235
0 A_3
Next, substituting x -1 to find A_2:
235 A_1 * (-1)^3 A_2 * (-1)^2 A_3 * (-1) 235
235 -A_1 A_2 0 235
0 -A_1 A_2
Given that A_3 0, we have:
0 -A_1 A_2
And since A_1 A_2 A_3 0 and A_3 0, we get:
0 A_1 A_2
From these equations, we can deduce that:
A_1 -A_2
Since we have no further information, we can assume A_1 0 and A_2 0 for simplicity. Therefore, the final decomposition is:
235 / (x^4) 235 / (x^4)
Or, more generally:
235 / (x^4) 0 / x 0 / x^2 0 / x^3 235 / x^4
Conclusion
Through the process of partial fraction decomposition, we have simplified the expression 235 / (x^4) into a form that is easier to work with. The key steps involved are: factoring the denominator, writing the expression as a sum of partial fractions, and determining the coefficients by substitution and algebraic manipulation. While the given expression simplified directly to the same form, the process remains a valuable technique for more complex rational functions.
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