Problem 67 - Algebra 2

Question

Simplify each rational expression. $$ \frac{x^{2}+4 x+3}{x^{2}-3 x-4} $$

Step-by-Step Solution

Answer

The simplified form of the given rational expression is \( \frac{x+3}{x-4}\)

Step 1: Factorize quadratic expressions

We need to factor the quadratic expressions in both numerator and denominator. Start with the numerator \(x^{2} + 4x + 3\). Look for two numbers that multiply to form 3 and add to form 4. We find 1 and 3 satisfy the condition, so we can factor the numerator as \( (x+1)(x+3)\). Follow the same procedure for the denominator \(x^{2} - 3x - 4\). Look for two numbers that multiply to form -4 and add to form -3. We find -4 and 1 satisfy these conditions. As a result, we can write the denominator as \( (x-4)(x+1)\) .

Step 2: Simplify the rational expression

We can simplify the expression by canceling common factors from the numerator and denominator. Both \(x+1\) are common factors. Therefore the expression simplifies to \( \frac{x+3}{x-4}\) .

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial skill in algebra. It's all about transforming a quadratic expression into a product of two binomials. Think of it as finding two numbers that fit together to replace a polynomial. To factor a quadratic expression like \(x^2 + 4x + 3\), follow these steps:

• Identify the numbers you need to find. Look for two numbers that multiply to give the constant term (3 in this case) and add up to the coefficient of the middle term (4).
• In this example, 1 and 3 fit the bill since \(1 \times 3 = 3\) and \(1 + 3 = 4\).
• Write the expression as \((x + 1)(x + 3)\).

For the denominator \(x^2 - 3x - 4\), apply the same strategy:

• You're looking for numbers that multiply to -4 and add up to -3.
• -4 and 1 work here, as \(-4 \times 1 = -4\) and \(-4 + 1 = -3\).
• Thus, the expression becomes \((x - 4)(x + 1)\).

By factoring quadratics, you set the stage for simplifying complex algebraic expressions.

Rational Expression

A rational expression is similar to a fraction, but with polynomials in both the numerator and the denominator. It's crucial to understand because it's common in algebra to work with these forms. Rational expressions can seem challenging, but breaking them down step-by-step makes them manageable.

• First, view the expression as a division problem between polynomials. For example, the expression given is \( \frac{x^2 + 4x + 3}{x^2 - 3x - 4} \).
• After factoring the quadratics, the expression becomes \( \frac{(x+1)(x+3)}{(x-4)(x+1)} \).

Rational expressions are manipulated by performing operations like addition, subtraction, multiplication, and division through a similar process to working with fractions. The key is always to keep everything balanced, just as you would when dealing with numbers in arithmetic. Expressing the rational expressions in their simplest form, like transforming our example into \( \frac{x+3}{x-4} \), is often the final goal.

Canceling Common Factors

Canceling common factors is the crucial step that simplifies rational expressions effectively. It's akin to reducing a fraction by removing common divisors. Once you've factored both the numerator and the denominator, as we've done with the expressions here, look for terms that appear in both.

• In our example, \((x+1)\) is a common factor in both the numerator and the denominator.
• Cancel the common factor out to simplify the expression. After canceling \((x+1)\), the rational expression \( \frac{(x+1)(x+3)}{(x-4)(x+1)} \) becomes \( \frac{x+3}{x-4} \).

Remember, you can only cancel factors, not terms, which means each part of the expression must be part of a product, not a sum or difference. This process reduces complexity and reveals the simplest form of the rational expression.