Problem 35 - Algebra 1

Question

Create three problems of the form \(\frac{a x^{2}+b x+c}{d x^{2}+e x+f}\) in which the numerator and the denominator have a common factor. Describe the process you used to create your problems.

Step-by-Step Solution

Answer

The three problems of the form \(\frac{a x^{2}+b x+c}{d x^{2}+e x+f}\) created, where the numerator and denominator have a common factor, are: \(\frac{x^2 - x - 2}{x^2 - 5x + 6}\), \(\frac{x^2 - 2x - 8}{x^2 + 5x + 6}\), and \(\frac{x^2 - 3x + 2}{x^2 + 3x - 4}\)

Step 1: Creating Problem 1

Start with a factor, for instance \(x - 2\). Multiply this factor by \(x + 1\) to get a polynomial for the numerator, \(x^2 - x - 2\). For the denominator, multiply the chosen factor by \(x - 3\), to get \(x^2 - 5x + 6\). This gives the problem: \(\frac{x^2 - x - 2}{x^2 - 5x + 6}\)

Step 2: Creating Problem 2

Choose a different factor, for instance \(x + 2\). Multiply this factor by \(x - 4\) to get a polynomial for the numerator, \(x^2 - 2x - 8\). For the denominator, multiply the chosen factor by \(x + 3\), to get \(x^2 +5x + 6\). This gives the problem: \(\frac{x^2 - 2x - 8}{x^2 + 5x + 6}\)

Step 3: Creating Problem 3

For the last problem, choose a factor like \(x - 1\). Multiply this factor by \(x - 2\) to get the polynomial for the numerator, \(x^2 - 3x + 2\). For the denominator, multiply the chosen factor by \(x + 4\), to get \(x^2 + 3x - 4\). This gives the problem: \(\frac{x^2 - 3x + 2}{x^2 + 3x - 4}\)

Key Concepts

Creating Polynomial Problems

Formulating polynomial problems that involve rational expressions can be an enriching exercise to understand the relationship between factors and the resultant polynomials. When creating problems where both the numerator and denominator are polynomials with a common factor, we essentially plant a 'hidden' simplification opportunity within the expression.

To engineer such problems, start by choosing a common factor for both the numerator and the denominator. This factor can be a simple binomial such as (x - a), where a is any real number. Next, each part of the fraction is formed by multiplying the common factor by another distinct polynomial, which could also be a binomial, creating two quadratic expressions. The common factor will be the key to simplifying the rational expression in later steps.

Here is a step-by-step approach:

• Select a common factor, such as (x - 2).
• Multiply this by a different binomial, like (x + 1), to create the numerator.
• Then, construct the denominator by multiplying the common factor by another binomial, for example, (x - 3).
• The resulting rational expression, (x^2 - x - 2)/(x^2 - 5x + 6), has the common factor seeded into both the numerator and denominator.

This technique allows students to practice and recognize the role factors play in the structure of polynomials within rational expressions.

Factoring Polynomials

Factoring polynomials is a critical skill in algebra that provides insight into a polynomial's properties and aids in solving equations. The process of factoring transforms a polynomial into a product of simpler polynomials or factors that, when multiplied together, yield the original polynomial.

Here's how factoring might look in practice:

• Consider a quadratic polynomial, such as x^2 + bx + c.
• Look for two numbers that multiply to c (the constant term) and add up to b (the coefficient of x).
• These two numbers can be used to break apart the middle term, leading to a factored form: (x + m)(x + n), where m and n are the numbers found.

Correctly identifying the factors prepares you to further simplify rational expressions where these factors may cancel out, provided they appear in both the numerator and the denominator.

In the context of rational expressions, factoring both the numerator and the denominator completely is pivotal, as this reveals any common factors that can then be simplified or canceled out.

Simplifying Rational Expressions

The goal of simplifying rational expressions is to reduce them to their simplest form by eliminating common factors in the numerator and the denominator. Simplifying isn't just about making an expression look 'neater'; it reveals the most fundamental form of the expression, often making it easier to evaluate or manipulate in equations and inequalities.

To simplify a rational expression:

• Factor both the numerator and the denominator completely.
• Identify any common factors shared by both.
• Divide out the common factors, effectively canceling them from both parts of the fraction.

For example, if you have a rational expression (x^2 - 4)/(x^2 - 4x + 4), factoring gives you ((x - 2)(x + 2))/(x - 2)^2. You can then cancel the common factor of (x - 2) to simplify it to (x + 2)/(x - 2).

This process of simplification is crucial for analyzing and understanding the behavior of rational expressions, especially concerning their domain, asymptotes, and intercepts. After simplifying, you're left with a rational expression stripped of redundancies and reflecting the true nature of the algebraic fraction.