Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the partial fraction decomposition for the rational expression. Check your result algebraically by combining fractions, and check your result graphically by using a graphing utility to graph the rational expression and the partial fractions in the same viewing window. $$\frac{x-2}{x^{2}+4 x+3}$$

Step-by-Step Solution

Answer

The partial fractions decompositions of the given rational expression \(\frac{x-2}{x^{2}+4 x+3}\) are \(-\frac{1}{x+1} + \frac{1}{x+3}\).

Step 1: Factorize the Denominator

First the denominator of the rational expression needs to be factored. The quadratic expression \(x^{2}+4x+3\) can be factored into \((x+1)(x+3)\). Therefore, the given fraction becomes \(\frac{x-2}{(x+1)(x+3)}\).

Step 2: Decompose into Partial Fractions

The general form for the partial fractions of the given expression is \(\frac{A}{x+1} + \frac{B}{x+3}\) where A and B are constants to be found. To find values of A and B, the two fractions can be combined and equated to the original fraction, which gives \((x-2) = A \cdot (x+3) + B \cdot (x+1)\). Equating the coefficients gives two simultaneous equations.

Step 3: Solve for Constants

On solving the two simultaneous equations from the previous step for A and B, we get A = -1 and B = 1. Therefore, the partial fractions of the given expression are \(-\frac{1}{x+1} + \frac{1}{x+3}\).

Step 4: Check Algebraically

To verify the result, combine the obtained fractions and see if it equals the original expression. We can verify that : \((-1)/(x+1) + 1/(x+3) = (x-2)/((x+1)(x+3))\).

Step 5: Check Graphically

To validate the result graphically, graph both the given rational expression function and the obtained function from the decomposition. If the decomposed function matches with the original function, then the solution is correct. The graphical representation does not provide a definite numerical value, but serves as a visual verification.

Key Concepts

Rational Expression

Rational expressions are like fractions, but instead of integers for the numerator and denominator, there are polynomials. In a simple fraction, the numerator is divided by the denominator to give a quotient, while in a rational expression, the division of polynomial functions is involved. For example, in the exercise, the rational expression is \( \frac{x-2}{x^2+4x+3} \). Here, \(x-2\) is the numerator and \(x^2+4x+3\) is the denominator.

This involves operations like addition, subtraction, multiplication, and division between polynomial expressions, just like you would with numbers. When dealing with rational expressions, simplifying and manipulating them often requires additional steps such as factoring and finding common denominators to achieve desired forms.

Quadratic Equation

A quadratic equation is a second-degree polynomial. It has the general form \(ax^2 + bx + c = 0\). The focus in our exercise is the fact that the denominator of the rational expression is a quadratic expression: \(x^2 + 4x + 3\).

Quadratic equations can have different forms:

• Standard form: \( ax^2 + bx + c = 0 \)
• Vertex form: \( a(x-h)^2 + k = 0 \)
• Factored form: \( a(x-p)(x-q) = 0 \), where \(p\) and \(q\) are the roots of the equation.

Solving quadratic equations typically involves factoring. However, other methods include using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), or completing the square. In this exercise, factorization was necessary to break down the rational expression.

Factoring Polynomials

Factoring polynomials is a method used to break down complicated polynomial expressions into simpler products. This is a crucial step in partial fraction decomposition as seen in this exercise where the quadratic expression \(x^2 + 4x + 3\) was factored into \((x+1)(x+3)\).

There are several strategies for factoring polynomials, such as:

• Identifying common factors in all terms.
• Using the quadratic technique for expressions with degree 2.
• Applying synthetic division or long division for higher-degree polynomials.

Successful factorization allows polynomials to be simplified and can make solving equations more manageable by breaking them down into linear forms.

Simultaneous Equations

Simultaneous equations are a set of equations with multiple variables that are solved together. In this context, they arise when the partial fraction decomposition requires finding unknown constants by equating coefficients.

From Step 2 of the solution process, the equation \( (x-2) = A(x+3) + B(x+1) \) leads to simultaneous equations when expanded:

• The coefficient of \(x\) gives: \( A + B = 1 \)
• The constant terms provide: \( 3A + B = -2 \)

By solving these equations, we determine the values of \(A\) and \(B\), which are \( A = -1 \) and \( B = 1 \). These solutions make it possible to decompose the original fraction into simpler parts.