Question

Resolve \(\frac{1}{x^{4}-3 x^{2}-4}\) into partial fractions. (1) \(\frac{1}{5\left(x^{2}-4\right)}+\frac{1}{5\left(x^{2}+1\right)}\) (2) \(\frac{1}{5\left(x^{2}-4\right)}-\frac{1}{5\left(x^{2}+1\right)}\) (3) \(\frac{1}{4\left(x^{2}+4\right)}-\frac{1}{4\left(x^{2}+1\right)}\) (4) \(\frac{1}{4\left(x^{2}+4\right)}+\frac{1}{4\left(x^{2}+1\right)}\)

Short answer

Based on the step-by-step solution provided above, decompose the rational function into partial fractions. Given function: $$\frac{1}{x^4-3x^2-4}$$ Partial fraction decomposition: $$\frac{1}{4\left(x^{2}+4\right)}-\frac{1}{4\left(x^{2}+1\right)}$$

Step-by-step solution

Identify denominators of potential partial fractions

The first step is identifying what the denominators of the partial fractions could be. To do that, the denominator polynomial, \(x^4-3x^2-4\), must be factored. Notice that it resembles a quadratic polynomial with \(x^2\) rather than \(x\). Therefore, perform a substitution \(y = x^2\): $$y^2-3y-4$$ Now, factor the quadratic polynomial: $$(y-4)(y+1)$$ Substitute \(x^2\) back for \(y\): $$(x^2-4)(x^2+1)$$ The potential partial fractions have denominators \((x^2-4)\) and \((x^2+1)\).

Set up the partial fractions

Set up the partial fraction decomposition of \(\frac{1}{x^4-3x^2-4}\) as follows (where A and B are constants to be found): $$\frac{1}{(x^2-4)(x^2+1)} = \frac{A}{x^2-4} + \frac{B}{x^2+1}$$ Now, we'll find the values of A and B.

Clear denominators and solve

Clear the denominators by multiplying the whole equation by \((x^2-4)(x^2+1)\) on both sides: $$1 = A(x^2+1) + B(x^2-4)$$ Now, we can solve for \(A\) and \(B\), by using the method of equating coefficients. The coefficients of \(x^2\) and the constant terms must be equal on both sides of the equation. $$1 = (A+B)x^2 - 4B$$ So, we have two equations: 1) \(A+B=0\) 2) \(-4B=1\) Solve the second equation for B: $$B=\frac{-1}{4}$$ Now, use the value of B to find A: $$A = -B = \frac{1}{4}$$

Combine the results

Put everything together to get the final decomposition in the form of the given options: $$\frac{1}{x^4-3x^2-4} = \frac{1/4}{x^2-4} + \frac{-1/4}{x^2+1}$$ This matches option (3), so the correct answer is: $$\frac{1}{4\left(x^{2}+4\right)}-\frac{1}{4\left(x^{2}+1\right)}$$

Key concepts

Polynomial Factorization

Partial fraction decomposition often involves factoring polynomials, especially the denominators of rational expressions. In the provided exercise, the denominator is a quartic polynomial: \(x^4-3x^2-4\). To simplify the process, we can treat this polynomial like a quadratic by substituting \(y = x^2\). This turns the original expression into \(y^2 - 3y - 4\), which is a quadratic polynomial. Quadratics are generally easier to factor using known techniques such as finding roots or applying the quadratic formula.
\[ (y - 4)(y + 1) \]After the substitution, factor the quadratic form as \((y-4)(y+1)\). Then replace \(y\) back with \(x^2\), obtaining the factored form \((x^2-4)(x^2+1)\). This step is crucial as it helps break down the fraction into simpler parts, which can then be used for partial fraction decomposition.

Rational Expressions

A rational expression is essentially a fraction where the numerator and the denominator are polynomials. In solving the exercise, we have a single rational expression \(\frac{1}{x^4-3x^2-4}\).
Recognizing that this can be decomposed into simpler fractions with linear or quadratic denominators is the key to resolving many problems in calculus and algebra. The essence of the exercise is to express this complex rational expression in terms of sums or differences of simpler fractions. By factoring the original polynomial denominator, you create the framework necessary for applying partial fraction techniques.

• Identify the polynomial in the denominator.
• Factor this polynomial to find potential denominators for your partial fractions.
• Express the complex rational expression as a sum or difference of simpler fractions whose denominators are these factors.

Gaining proficiency with these ideas aids in understanding and solving differential equations, integrating rational functions, or working with series expansions.

Algebraic Manipulation

Once your polynomial is factored and potential partial fractions identified, algebraic manipulation comes into play. This manipulation involves setting up equations and solving for unknown constants. The rational expression \(\frac{1}{(x^2-4)(x^2+1)}\) is decomposed into partial fractions:
\[\frac{A}{x^2-4} + \frac{B}{x^2+1}\]To identify values of \(A\) and \(B\), clear the denominators by multiplying through by the common denominator and then simplify the resulting equation:
\[1 = A(x^2+1) + B(x^2-4)\]Equate coefficients from both sides of the equation to form a system of linear equations:

• \(A+B = 0\)
• \(-4B = 1\)

Solving these yields \(B = -\frac{1}{4}\) and \(A = \frac{1}{4}\). Combining these constants back into the partial fractions gives the final expression that matches one of the exercise options, illustrating how algebraic manipulation is crucial for finding solutions in polynomial factorizations and rational expressions.