Problem 19 - A Graphical Approach to Precalculus with Limits

Question

Find the partial fraction decomposition for each rational expression. $$\frac{x^{2}}{x^{2}+2 x+1}$$

Step-by-Step Solution

Answer

\( \frac{1}{x+1} - \frac{1}{(x+1)^{2}} \)

Step 1: Recognize the Denominator

First, we need to identify the factorization of the denominator \( x^{2} + 2x + 1 \). Notice that this is a perfect square trinomial, which can be factored into \((x + 1)^{2}\) because \((x + 1)(x + 1) = x^{2} + 2x + 1\).

Step 2: Setup the Partial Fraction

Given the denominator \((x + 1)^2\), the expression should be decomposed into terms involving \( (x + 1) \) and \( (x + 1)^2 \). The standard form for the partial fraction decomposition is \[ \frac{A}{x+1} + \frac{B}{(x+1)^{2}}. \]

Step 3: Combine the Fractions

Combine the two fractions from Step 2 over a common denominator. This involves writing: \[ \frac{A(x+1) + B}{(x+1)^2} = \frac{x^{2}}{(x+1)^2}. \] This equation must hold for all values of \(x\).

Step 4: Equate the Numerators

Since the denominators are equal, equate the numerators: \( A(x + 1) + B = x^2 \). Simplify the left side: \( Ax + A + B = x^2 \).

Step 5: Solve for Coefficients

Now, match the coefficients of like terms from both sides of the equation. Equating the coefficients gives us the system \[ A = 1, \quad A + B = 0. \] From \( A = 1 \), substitute into the second equation: \( 1 + B = 0 \). Solving for \( B \) gives \( B = -1 \).

Step 6: Write the Decomposition

Substitute the solved coefficients back into the partial fraction expression: \[ \frac{x^{2}}{x^{2}+2x+1} = \frac{1}{x+1} - \frac{1}{(x+1)^{2}}. \]

Key Concepts

Rational Expression

A rational expression is like a fraction, but instead of just numbers, it involves variables. It's in the form of \( \frac{a(x)}{b(x)} \), where both the numerator \( a(x) \) and the denominator \( b(x) \) are polynomials. In the exercise given, the rational expression is \( \frac{x^2}{x^2+2x+1} \). The numerator, \( x^2 \), is a simple polynomial consisting of a single term. The denominator, \( x^2+2x+1 \), is more complex as it consists of several terms.Understanding rational expressions is crucial because they appear frequently in algebra and calculus. They can represent complex relationships and are often manipulated by factoring, simplifying, or decomposing them as in this exercise. Managing these expressions involves:

• Identifying the polynomial in the numerator and denominator.
• Simplifying the fraction by factoring and canceling common factors when possible.
• Using techniques like partial fraction decomposition to break down complex expressions into simpler parts.

Grasping rational expressions opens the door to solving more advanced mathematical problems and can enhance your skills in equation manipulation.

Perfect Square Trinomial

A perfect square trinomial is a unique type of polynomial with three terms. It can be written as the square of a binomial. In fact, identifying and working with perfect square trinomials is a fundamental skill in algebra.For our exercise, the denominator \( x^2 + 2x + 1 \) is a perfect square trinomial. It can be rewritten as \((x + 1)^2\). This happens because when you expand \((x + 1)(x + 1)\), using the distributive property or FOIL method, you get back \(x^2 + 2x + 1\).Recognizing perfect square trinomials allows for simpler expression manipulation since:

• They can be factored neatly into binomial squares.
• This factorization often helps in simplifying and solving algebraic equations.
• It lays the foundation for understanding quadratic equations and their solutions.

By mastering the identification of perfect square trinomials, you can easily spot patterns and apply this knowledge, aiding in more advanced algebraic tasks.

Denominator Factorization

Denominator factorization is the process of breaking down a complex polynomial in a denominator into simpler, more manageable factors. This simplification is vital when dealing with rational expressions, especially when you're aiming to perform operations like partial fraction decomposition.In the given exercise, the denominator \( x^2 + 2x + 1 \) factors into \((x + 1)^2\) easily because it's a perfect square trinomial. Proper factorization arms you with the ability to:

• Identify repeated factors or roots, which is necessary for partial fractions.
• Simplify the algebraic expression by canceling out common factors with the numerator if possible.
• Make complex algebraic operations more straightforward, enhancing problem-solving efficiency.

Mastering denominator factorization will significantly aid in evaluating, simplifying, and decomposing rational expressions in algebra, making it a key competency in your math toolkit.