Question

Decompose \(\left(\mathrm{x}^{3}+5 \mathrm{x}-4\right) /\left(\mathrm{x}^{2}-\mathrm{x}-2\right)\) into partial fractions.

Short answer

The decomposition of the given rational function into partial fractions is: \[ \frac{x^3 + 5x - 4}{x^2 - x - 2} = x + \frac{14/3}{x - 2} - \frac{10/3}{x + 1} \]

Step-by-step solution

Perform Polynomial Division

First, we need to perform polynomial division to divide the given expression: \[ \frac{x^3 + 5x - 4}{x^2 - x - 2} \] The long division process is as follows: ``` x ------------------- x^2 - x - 2 | x^3 + 0x^2 + 5x - 4 x^3 - x^2 - 2x --------------- x^2 + 7x - 4 x^2 - x - 2 ---------- 8x - 2 ``` Thus, the division results are: Quotient: \(x\) Remainder: \(8x - 2\) So, the expression can be rewritten as: \[ x + \frac{8x - 2}{x^2 - x - 2} \] Now, we will perform partial fraction decomposition for the remaining fraction \( \frac{8x - 2}{x^2 - x - 2} \).

Factor the Denominator

We need to factor the denominator of the fraction: \[ x^2 - x - 2 = (x - 2)(x + 1) \]

Write Partial Fraction Decomposition

Now, we will rewrite the rational function as a sum of partial fractions: \[ \frac{8x - 2}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1} \]

Clear Fractions

Now clear the fractions by multiplying both sides of the equation by the common denominator \((x - 2)(x + 1)\): \[ 8x - 2 = A(x + 1) + B(x - 2) \]

Solve for A and B

Now, we will solve for \(A\) and \(B\) by choosing suitable values of \(x\) which make one term zero: 1. Choose \(x = 2\): \[ 8(2) - 2 = A(2 + 1) \\ 14 = 3A \\ A = \frac{14}{3} \] 2. Choose \(x = -1\): \[ 8(-1) - 2 = -B(-1 - 2) \\ -10 = 3B \\ B = -\frac{10}{3} \]

Write the Final Partial Fractions

Now substitute the values of A and B into the partial fractions: \[ \frac{8x - 2}{(x - 2)(x + 1)} = \frac{14/3}{x - 2} - \frac{10/3}{x + 1} \]

Write the Final Answer

Don't forget to include the quotient from the polynomial division at the beginning: \[ \frac{x^3 + 5x - 4}{x^2 - x - 2} = x + \frac{14/3}{x - 2} - \frac{10/3}{x + 1} \] So, the decomposition of the given rational function into partial fractions is: \[ x + \frac{14/3}{x - 2} - \frac{10/3}{x + 1} \]

Key concepts

Polynomial Division

Polynomial division is an essential technique when dealing with rational functions, particularly when simplifying complex expressions or converting them into partial fractions. The process closely resembles traditional long division used with numbers but is applied to polynomial expressions. In our exercise, we start by dividing the polynomial in the numerator, which is of a higher degree, by the polynomial in the denominator.

• First, identify the leading terms in both the numerator and the denominator.
• Divide the leading term of the numerator by the leading term of the denominator to get the first term of the quotient.
• Multiply the entire divisor (the polynomial in the denominator) by this term of the quotient and subtract the result from the numerator.
• Repeat the process with the new polynomial formed from the difference until the remainder has a lower degree than the divisor.

This process reexpresses the original rational function as a polynomial (quotient) plus a new rational function, where the degree of the numerator is less than that of the denominator. It's this remainder fraction that we will further decompose into partial fractions.

Factorization

Factorization is the key step that allows us to express the remainder of our divided rational function as partial fractions. When we factor, we are looking to rewrite an expression as a product of its simpler components.
The denominator of our fraction is a quadratic expression, which often can be factored into two linear parts - when the quadratic is factorable using real numbers.For our specific quadratic \(x^2 - x - 2\), the steps are:

• Find two numbers that multiply to give the constant term (-2) and add up to the coefficient of the linear term (-1).
• These numbers are -2 and 1.
• Rewrite the expression as \((x - 2)(x + 1)\).

By factorizing the denominator, we set the stage to separate the rational expression into simpler components, each having their distinct expressions in partial fraction decomposition. Factorization transforms our remaining problem from polynomial division into a more manageable form.

System of Equations

Once we have expressed our fraction in terms of its factorized form in the denominator, we proceed to set up a system of equations. By writing the partial fraction decomposition as:\[\frac{8x - 2}{(x - 2)(x + 1)} = \frac{A}{x - 2} + \frac{B}{x + 1}\]Multiply every term by the common denominator to clear the fractions: \( (x - 2)(x + 1) \). This leads to an equation without denominators:\[8x - 2 = A(x + 1) + B(x - 2)\]To solve for the constants \(A\) and \(B\), choose strategic values for \(x\) that simplify the equation:

• Set \(x = 2\) to nullify the \(B\)-term, solving for \(A\).
• Set \(x = -1\) to nullify the \(A\)-term, solving for \(B\).

The system of equations approach provides us with the values of the unknowns, \(A\) and \(B\), completing the partial fraction decomposition process.

Rational Functions

Rational functions are expressions that are the ratio of two polynomials. They often appear in mathematical analysis and have unique properties and ways of manipulation, particularly using partial fraction decomposition.In our exercise, the rational function is \(\frac{x^3 + 5x - 4}{x^2 - x - 2}\).The aim with rational functions is often to simplify expressions, solve equations or integrate them more easily by breaking them into simpler fractions. Partial fraction decomposition is a powerful tool used for these tasks, particularly useful in calculus for integrating rational functions.The decomposition allows us to express the function as a sum of simpler terms that are easier to handle or interpret:

• Gaining a clearer picture of the behavior of the function across its domain.
• Identifying asymptotes which indicate the values that the function approaches but never reaches.
• Simplifying the process of integration by replacing complex rational functions with simpler, indeed elementary, fractions.

Understanding and decomposing rational functions ultimately help in further exploring areas like differential equations, control theory, and systems analysis.