Problem 11 - Calculus Early Transcendentals

Question

Give the partial fraction decomposition for the following functions. $$\frac{x+2}{x^{3}-3 x^{2}+2 x}$$

Step-by-Step Solution

Answer

Answer: The partial fraction decomposition of the given function is $$\frac{x+2}{x^{3}-3 x^{2}+2 x} = \frac{-1}{x} - \frac{3}{x-1} + \frac{2}{x-2}$$.

Step 1: Factor the Denominator

Factor the denominator to simplify the expression and identify the individual fractions. Denominator: \(x^{3}-3 x^{2}+2 x = x(x^{2}-3x+2) = x(x-1)(x-2)\). The factored denominator is now \(x(x-1)(x-2)\).

Step 2: Set up Partial Fraction Decomposition

Now, set up the partial fraction decomposition using the factored denominator: $$\frac{x+2}{x(x-1)(x-2)} = \frac{A}{x} + \frac{B}{x-1} + \frac{C}{x-2}$$

Step 3: Clear the Denominator

Multiply both sides by the denominator to clear it: $$x+2 = A(x-1)(x-2) + Bx(x-2) + Cx(x-1)$$

Step 4: Solve for A, B, and C

To find the values for A, B, and C, we can substitute values of x that simplify the equation: For A: Let \(x=0\), and the equation simplifies to: $$2 = -A(1)(-2) \implies A = -1$$ For B: Let \(x=1\), and the equation simplifies to: $$3 = B(1)(-1) \implies B = -3$$ For C: Let \(x=2\), and the equation simplifies to: $$4 = C(2)(1) \implies C = 2$$

Step 5: Write the Partial Fraction Decomposition

Substitute the obtained values of A, B, and C back into the partial fraction decomposition: $$\frac{x+2}{x(x-1)(x-2)} = \frac{-1}{x} + \frac{-3}{x-1} + \frac{2}{x-2}$$ The partial fraction decomposition of the given function is: $$\frac{x+2}{x^{3}-3 x^{2}+2 x} = \frac{-1}{x} - \frac{3}{x-1} + \frac{2}{x-2}$$

Key Concepts

Factoring Polynomials

Factoring polynomials is the process of breaking down a complex polynomial into simpler terms or factors that, when multiplied together, give you the original polynomial. This step is critical for simplifying expressions, particularly in partial fraction decomposition. To factor a polynomial effectively:

• Identify and take out the greatest common factor, if any.
• Look for patterns such as difference of squares, perfect square trinomials, or the sum/difference of cubes.
• Consider using techniques such as grouping or splitting the middle term for more complicated polynomials.

In our exercise, the polynomial in the denominator is simplified by first taking out the common factor 'x', and then factorizing the quadratic expression \(x^2 - 3x + 2\), which is solved into \((x-1)(x-2)\). This gives us the complete factorization \(x(x-1)(x-2)\). Factoring polynomials efficiently is crucial for simplifying mathematical expressions and solving equations later on.

Solving Equations

Solving equations is about finding the value(s) of the variable(s) that make an equation true. In partial fraction decomposition, after expressing the fraction as a sum of simpler parts, next is to solve for the unknown coefficients. Here’s how we approach it:

• After setting up the partial fraction form, clear the denominators by multiplying through by the original denominator.
• Substitute strategic values of \(x\) that simplify one side of the equation, helping isolate terms with specific unknown coefficients.
• Substituting values like the roots of the factored terms often zero out certain coefficients, making it simpler to solve for others directly.

In this case, substituting \(x = 0\), \(x = 1\), and \(x = 2\) simplified our equations and helped us solve for \(A\), \(B\), and \(C\) easily. This method, knowing what values will simplify your equations, and isolating unknowns are key skills in solving algebraic equations.

Rational Expressions

Rational expressions are fractions in which the numerator and/or the denominator are polynomials. Simplifying these expressions often involves factoring both the numerator and the denominator, canceling common factors, and sometimes using partial fraction decomposition.Understanding the behavior of rational expressions is essential as they can represent complex relationships. When working with these:

• Always check for factors common to the numerator and the denominator.
• Be aware of restrictions on the variable that make the denominator zero, as these cause the expression to be undefined.
• Use long division, if needed, when the degree of the numerator is not less than the degree of the denominator before you can apply partial fraction techniques.

The exercise demonstrates these principles by first simplifying the expression \(\frac{x+2}{x^3-3x^2+2x}\) into manageable parts. Once factored, it illustrates how each part corresponds to a simpler rational expression, which when summed, return the original expression. Mastering rational expressions is hence very important in advanced algebra topics, especially when manipulating algebraic fractions.