Problem 65 - College Algebra Essentials

Question

Find the partial fraction decomposition of $$ \frac{4 x^{2}+5 x-9}{x^{3}-6 x-9} $$

Step-by-Step Solution

Answer

The solution steps provided here in detail require careful working and accuracy to get the right constants in the partial fraction. Once found, these are substituted back to get the final expression which forms the partial fraction decomposition. The specific constants and thus the final solution would depend on the specific roots of the denominator.

Step 1: Check the degree of polynomial

Check if the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator. In this case it is so \( 2 < 3 \). Therefore, no polynomial division is needed. If the degree of the numerator was greater, perform polynomial division first.

Step 2: Factorize the denominator

If the degree of the polynomial in the denominator is two or greater, factorize it. This can be a challenging step which might require different approaches depending on the complexity of the denominator. In this case, the denominator does not factorize easily over the Real numbers. However, the denominator already expresses the denominator as a product of factors over the Complex numbers: \( (x - a)(x - b)(x - c) = x^{3}-6 x-9 \). As this function has 3 roots, it can be written as \( A/ (x - a) + B/ (x - b) + C/ (x - c) \), where \( a, b, c \) are the roots of the denominator and \( A, B,C \) are constants to be determined.

Step 3: Determine the constants

Multiply through by the denominator \( (x - a)(x - b)(x - c) \) to clear fractions to get an equation that depends only on \( x \). This equation is then used to find the values of constants \( A, B, C \). The constant A can be found by substituting root a of the denominator in this equation. Similar method is applied to find B,C. Note that this process can be long and requires careful working to avoid errors.

Step 4: Write down the final expression

Once the constants A, B and C are determined, they are substituted back into the expression to get the partial fraction decomposition of the original function.

Key Concepts

Polynomial Division

When we encounter a rational expression where the numerator has a higher or equal degree compared to the denominator, the first step is to perform polynomial division. This is similar to long division with numbers but involves polynomials instead.

Polynomial division aims to simplify the expression so that the degree of the numerator becomes less than the degree of the denominator. This process often results in a polynomial (the quotient) and possibly a remainder. The quotient is written separately, and the remainder is expressed over the original denominator. When working through polynomial division, we can apply the concept of dividing each term of the numerator, by the highest degree term of the denominator, subtracting the resulting expression from our original polynomial, and repeating the process until the remainder's degree is less than the denominator's degree.

If the numerator's degree is already lower, as in the given exercise, polynomial division isn't necessary, and we can proceed directly to the next step.

Factorization of Polynomials

Factorization is the process of breaking down a complicated polynomial into simpler polynomials that are multiplied together. Factors are typically easier to work with, especially when solving equations or simplifying expressions.

Polynomial factorization can be accomplished using several methods, including factoring by grouping, using the distributive property (reverse of expanding), applying formulas for special products, or even by employing the Rational Roots Theorem for polynomials with integer coefficients. However, when the polynomial does not factorize nicely into real numbers, we may need to find its roots in the set of complex numbers, which can involve using techniques like completing the square or leveraging the quadratic formula.

Roots of Polynomial

The roots of a polynomial are the solutions to the equation formed by setting the polynomial equal to zero. In other words, they are the values of the variable that make the polynomial vanish. There can be as many roots as the degree of the polynomial, reflecting the Fundamental Theorem of Algebra.

In the case of real-valued polynomials, these roots can be real numbers or complex numbers. To identify them, various strategies might be employed based on the degree of the polynomial and its specific nature. Once the roots are found, as in the step-by-step solution provided in the exercise, we can rewrite the denominator of our rational expression as a product of linear factors corresponding to these roots.

Complex Numbers

Complex numbers are an extension of the real numbers and include all possible numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit, which is defined by the property that \( i^2 = -1 \).

The concept of complex numbers is fundamental in mathematics because they provide a complete field where every non-constant polynomial equation has a solution, as postulated by the Fundamental Theorem of Algebra. Complex numbers are particularly useful when dealing with polynomial roots that cannot be expressed with real numbers alone. These can include the roots of negative numbers and provide a crucial tool for solving polynomials of higher degrees, where some or all of the roots might not be real. Understanding how to work with complex numbers is essential to mastering techniques such as partial fraction decomposition of polynomials with complex roots.