Problem 30 - College Algebra

Question

Write the partial fraction decomposition of each rational expression. $$\frac{5 x^{2}-9 x+19}{(x-4)\left(x^{2}+5\right)}$$

Step-by-Step Solution

Answer

The partial fraction decomposition of the rational expression \(\frac{5x^{2}-9x+19}{(x-4)(x^{2}+5)}\) is \(\frac{A}{x-4} + \frac{Bx+C}{x^{2}+5}\), where \(A\), \(B\), and \(C\) are constants determined by comparing coefficients from both sides of the equation in step 3.

Step 1: Identify & Write the General Form

Identify the denominator's factors and construct the general form of the partial fraction decomposition.\nThe factors are \((x-4)\) and \((x^{2}+5)\).\nSince \((x^{2}+5)\) is an irreducible quadratic term, put it as \(Ax+B\) in numerator. So, the general form of the partial fraction decomposition will be:\n\(\frac{5x^{2}-9x+19}{(x-4)(x^{2}+5)} = \frac{A}{x-4} + \frac{Bx+C}{x^{2}+5}\).

Step 2: Clear Fraction and Collect Like Terms

Clear the fraction by multiplying both sides by the denominator of the left side to get rid of the complex fraction. Then, collect like terms.\nSo, this leads to the equation: \(5x^{2}-9x+19 = A(x^{2}+5) + Bx(x-4) + C(x-4)\).

Step 3: Solve for Coefficients

Next, solve for the values of \(A\), \(B\), and \(C\). This can be done by comparing the coefficients separately for \(x^{2}\), \(x\), and the independent term from both sides of the equation.

Step 4: Substitute values into the General Form

After finding the values of \(A\), \(B\), and \(C\), substitute these into the general form, you established in Step 1.

Key Concepts

Rational Expressions

Rational expressions are fractions wherein both the numerator and the denominator consist of polynomials. Understanding rational expressions is vital because they appear frequently in algebra and calculus.
A general rational expression looks like \(\frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials, and \(Q(x)\) cannot be zero. These expressions can often be simplified or rewritten using partial fraction decomposition.

In the context of partial fractions, rational expressions are broken down into simpler parts. This involves expressing a complex fraction as a sum of simpler fractions. For example, the rational expression \(\frac{5x^2-9x+19}{(x-4)(x^2+5)}\) can be decomposed into simpler fractions. This process helps solve equations and is particularly useful in integral calculus where integration of simpler fractions is easier than complex ones.

Quadratic Factors

Quadratic factors are polynomials of degree two and play a crucial role in partial fraction decomposition. Let's dissect what makes them special.

A quadratic factor is typically in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. If you can't further factor the quadratic factor over the real numbers, you retain it as it is in decompositions. For example, \(x^2+5\) in the expression \(\frac{5x^{2}-9x+19}{(x-4)(x^2+5)}\) cannot be factored further using real numbers.

• This means the partial fraction involving \(x^2+5\) will have a linear term \(Bx + C\) in the numerator, as each irreducible quadratic factor requires a linear numerator.

Including such quadratic factors in decompositions helps maintain the integrity of an expression, paving the way for accurate and effective solving whether in differentiation or integration tasks.

Irreducible Factors

An irreducible factor is a polynomial that cannot be factored further into polynomials of lower degrees using real number coefficients. Recognizing these is important when performing partial fraction decomposition.

In the rational expression \(\frac{5x^2-9x+19}{(x-4)(x^2+5)}\), \(x^2+5\) is an example of an irreducible quadratic factor since it cannot be simplified further in terms of real numbers. When encountering irreducible quadratic terms in the partial fraction process, use the structure \(Ax + B\) for the numerator over the irreducible factor.

• This ensures each decomposed part of your fraction respects the complexity of the factor it accompanies.

By understanding and properly handling irreducible factors, one can neatly and effectively decompose complex rational expressions for further mathematical manipulation.