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Chapter 9: Problem 33

Find the partial fraction decomposition of the given rational expression. $$ \frac{40}{x^{3}+3 x^{2}-4 x-12} $$

Short Answer

Expert verified
\(\frac{5}{x+3} + \frac{3}{x-2} - \frac{8}{x+2}\) is the partial fraction decomposition.

Step by step solution

01

Factor the Denominator

First, factor the cubic polynomial in the denominator: \[x^3 + 3x^2 - 4x - 12.\]Try factoring by grouping or synthetic division. Through testing potential rational roots (using the Rational Root Theorem), find that \(x + 3\) divides the polynomial. Use synthetic division to divide:\((x^3 + 3x^2 - 4x - 12) \div (x + 3) = (x^2 - 4).\)The factored form is:\[x^3 + 3x^2 - 4x - 12 = (x + 3)(x^2 - 4).\]
02

Factor the Quadratic Term

Next, factor the quadratic term \(x^2 - 4\), which is a difference of squares:\[x^2 - 4 = (x - 2)(x + 2).\]
03

Write Partial Fraction Decomposition

The denominator is now fully factored:\[(x+3)(x-2)(x+2).\]The partial fraction decomposition will have the form:\[\frac{A}{x+3} + \frac{B}{x-2} + \frac{C}{x+2}.\]
04

Set Up Equation and Solve for Constants

Multiply through by the common denominator \((x+3)(x-2)(x+2)\) to clear fractions:\[40 = A(x-2)(x+2) + B(x+3)(x+2) + C(x+3)(x-2).\]Expand and simplify this equation, and equate coefficients or substitute suitable values for \(x\) to solve for \(A\), \(B\), and \(C\).
05

Substitute Values of Constants

By substituting values:- Let \(x = -3\) gives \(40 = C(-3-2)(-3+2)\) thus \(C = -8.\)- Let \(x = 2\) gives \(40 = A(2+3)(2-2)\), thus \(A = 5.\)- Let \(x = -2\) gives \(40 = B(-2+3)(-2+2)\), thus \(B = 3.\)Now substitute these back into the partial fractions.
06

Write the Final Decomposition

Substitute the values of \(A\), \(B\), and \(C\) into the partial fraction expression:\[\frac{40}{x^3 + 3x^2 - 4x - 12} = \frac{5}{x+3} + \frac{3}{x-2} + \frac{-8}{x+2}.\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Rational Expressions
Rational expressions are essentially fractions that consist of two polynomials: one in the numerator and the other in the denominator. Understanding rational expressions is key to mastering partial fraction decomposition. Let's break it down further:
  • The numerator and denominator of a rational expression are polynomials. This means they can include variables raised to different powers, such as \(x^3\) or \(x^2\).
  • The goal is to express complex rational expressions as simpler parts that are easier to work with, known as partial fractions.
  • In the given exercise, the rational expression \(\frac{40}{x^{3}+3x^{2}-4x-12}\) is simplified by factoring the polynomial in the denominator. This allows you to break down a single complex expression into several simpler fractions.
Rational expressions often require techniques like factoring the denominator to simplify the expression to partial fractions, making it easier to integrate or perform other mathematical operations.
Synthetic Division
Synthetic division is a simplified form of polynomial division, especially useful when dividing by linear factors. In the given problem, synthetic division is used to simplify the polynomial denominator. Here鈥檚 how it works:

  • Identify the divisor, which in this case is \(x+3\). This linear factor tells us how we will divide the polynomial.
  • Using synthetic division, test roots that satisfy the original polynomial equation. Choose potential roots using the Rational Root Theorem.
  • The division helps to simplify the cubic polynomial \(x^3 + 3x^2 - 4x - 12\) into \((x + 3)(x^2 - 4)\).
Synthetic division is a quicker way to divide when compared to long division. It efficiently reduces the polynomial, providing a gateway to factored terms. This is crucial for partial fraction decomposition as it clarifies the structure, allowing us to work with simpler expressions.
Difference of Squares
The difference of squares is a mathematical identity that is used to factor expressions of the form \(a^2 - b^2\). It simplifies expressions into multiplicative factors. Here's why it's important:

  • For the exercise, the quadratic term \(x^2 - 4\) in the denominator can be seen as a difference of squares since \(4 = 2^2\).
  • Accordingly, \(x^2 - 4\) is factored as \((x - 2)(x + 2)\), simplifying the quadratic expression.
  • This factoring method is straightforward and provides deeper insights into the structure of polynomials, aiding in partial Fraction Decomposition.
Understanding and applying the difference of squares helps to swiftly break down polynomials into simpler multiplicative components. This is pivotal when preparing a denominator for partial fractions, leading to manageable pieces for computation.

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Most popular questions from this chapter

Solve the given linear system. State whether the system is consistent, with independent or dependent equations, or whether it is inconsistent. $$ \left\\{\begin{array}{r} 7 x-3 y-14=0 \\ x+y-1=0 \end{array}\right. $$

Find the partial fraction decomposition of the given rational expression. $$ \frac{t+8}{t^{4}-1} $$

Solve the given nonlinear system. $$ \left\\{\begin{array}{l} y=x\left(x^{2}-6 x+8\right) \\ y+4=(x-2)^{2} \end{array}\right. $$

Solve the given nonlinear system. $$ \left\\{\begin{array}{l} -2 x+\lambda=0 \\ y-y \lambda=0 \\ v^{2}-x=0 \end{array}\right. $$

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