Problem 10 - Precalculus

Question

(a) factor the denominator of the given \(\mathrm{ra}\) tional expression; (b) determine the form of the partial fraction decomposition for the given rational expression; and (c) determine the values of the constants in the partial fraction decomposition that you gave in part (b). To help you in spotting errors, use the fact that in part (c), each of the required constants turns out to be an integer. $$\frac{19 x-15}{4 x^{2}-5 x}$$

Step-by-Step Solution

Answer

\(\frac{19x-15}{x(4x-5)} = \frac{3}{x} + \frac{7}{4x-5}\) with constants \(A = 3\) and \(B = 7\).

Step 1: Factor the Denominator

First, identify the denominator to factor: \(4x^2 - 5x\). Look for a common factor in the polynomial. Here, the common factor is \(x\), so we can factor it out:\[4x^2 - 5x = x(4x - 5)\]

Step 2: Set Up Partial Fraction Decomposition

The expression can be decomposed since the degree of the numerator \(19x - 15\) is less than the degree of the denominator \(4x^2 - 5x\). Use the factored denominator, \(x(4x - 5)\), to write:\[\frac{19x-15}{x(4x - 5)} = \frac{A}{x} + \frac{B}{4x - 5}\]where \(A\) and \(B\) are constants to be determined.

Step 3: Equate and Simplify

To find \(A\) and \(B\), multiply through by the common denominator, \(x(4x - 5)\), to clear the fractions:\[19x - 15 = A(4x - 5) + Bx\]Expand both terms:\[19x - 15 = 4Ax - 5A + Bx\]Combine like terms:\[19x - 15 = (4A + B)x - 5A\]

Step 4: Solve the System of Equations

From the equation:\[19x - 15 = (4A + B)x - 5A\]we equate coefficients for \(x\) and the constant term:\[4A + B = 19\]\[-5A = -15\]Solve the second equation for \(A\):\[A = 3\]Substitute \(A = 3\) into the first equation:\[4(3) + B = 19\]\[12 + B = 19\]\[B = 7\]

Step 5: Finalize the Partial Fraction Decomposition

Now, substitute \(A = 3\) and \(B = 7\) back into the partial fraction decomposition:\[\frac{19x-15}{x(4x-5)} = \frac{3}{x} + \frac{7}{4x-5}\]

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra that involves breaking down a complex expression into simpler "factors" that can be multiplied together to obtain the original polynomial. In this exercise, we start with the polynomial in the denominator:
\[4x^2 - 5x\].To factor this polynomial, we first look for a common factor in the terms. Here, both terms have an \(x\) in common, so we factor \(x\) out, leading to:

• \[x(4x - 5)\]

Factoring is useful because it simplifies expressions and is an essential step before performing operations like Partial Fraction Decomposition. Recognizing factoring opportunities can greatly aid in solving algebraic problems efficiently.
Remember, factoring polynomials allows us to express them in terms of simpler terms which can be used to find the roots or simplify algebraic expressions.

Rational Expressions

Rational expressions are expressions that consist of a ratio of two polynomials. Similar to rational numbers being a quotient of two integers, rational expressions involve a numerator and a denominator, with the denominator not equal to zero. In this exercise, our rational expression is:

• \[\frac{19x - 15}{4x^2 - 5x}\]

The concept of Partial Fraction Decomposition comes into play here because it allows us to express a complex rational expression as a sum of simpler fractions. This is especially useful when integrating expressions or solving equations.
The process begins by ensuring that the polynomial in the numerator is of a lesser degree than that in the denominator, which is satisfied here. Following this, we use the factored form

• \[x(4x-5)\]

to set up our partial fractions:

• \[\frac{A}{x} + \frac{B}{4x - 5}\]

This breakdown simplifies integration and solves for constants "A" and "B" that can reassemble the original rational expression.

System of Equations

A system of equations is a set of equations with multiple variables that you solve simultaneously. In this partial fraction decomposition exercise, we have extracted constants \(A\) and \(B\) by solving a system of linear equations.The original equation after setting up the partial fractions is:

• \[19x - 15 = A(4x - 5) + Bx\]

By expanding and combining like terms, we obtain:

• \[(4A + B)x - 5A = 19x - 15\]

From here, we generate equations by equating coefficients of like terms:

• \[4A + B = 19\]
• \[-5A = -15\]

Solving these gives us values \(A = 3\) and \(B = 7\).
Systems of equations are vital as they allow connection between different equations to solve for unknowns, especially when handling more complex algebraic manipulations like partial fraction decomposition. These skills lay down the foundation for algebraic proficiency that can be applied in calculus and beyond.