Problem 45 - Intermediate Algebra

Question

Simplify each rational expression. $$ \frac{x^{2}+2 x+1}{x^{2}+4 x+3} $$

Step-by-Step Solution

Answer

\( \frac{x+1}{x+3} \)

Step 1: Identify and Factorize the Numerator

The numerator is \( x^2 + 2x + 1 \). This is a perfect square trinomial and can be factorized as \( (x+1)^2 \).

Step 2: Identify and Factorize the Denominator

The denominator is \( x^2 + 4x + 3 \). This quadratic expression can be factorized by finding two numbers that multiply to 3 and add to 4. These numbers are 1 and 3, so the factorization is \( (x+1)(x+3) \).

Step 3: Simplify the Expression

The original expression \( \frac{x^2 + 2x + 1}{x^2 + 4x + 3} \) simplifies to \( \frac{(x+1)^2}{(x+1)(x+3)} \). Cancel out the common factor \( (x+1) \) from the numerator and the denominator, leaving \( \frac{x+1}{x+3} \).

Key Concepts

Factoring Trinomials

Factoring trinomials is an essential skill in algebra, especially when working with rational expressions, quadratic equations, and polynomials. A trinomial is a polynomial with three terms, and factoring involves breaking it down into products of binomials or other polynomial forms. For instance, consider the trinomial \( x^2 + 2x + 1 \). This is a special type known as a perfect square trinomial because it can be written as \( (x+1)^2 \), where the middle term's coefficient (2) is twice the product of \( x \) and 1. To factor a trinomial, look for patterns or use methods like:

• Inspecting common factors: Look for common terms across the running terms.
• Recognizing special products: Identify if it matches forms like \( (a+b)^2 \) or \( (a-b)^2 \).
• Using trial and error: Find two numbers that multiply into the last term and add/subtract to the middle term's coefficient.

By familiarizing yourself with these techniques, factoring becomes less daunting and makes simplifying expressions and solving equations more manageable.

Simplifying Expressions

Simplifying expressions is a crucial process in algebra to make complex mathematical formulas easier to work with. This involves reducing expressions to their simplest form by canceling common factors in fractions, combining like terms, or factoring.When simplifying the rational expression \( \frac{x^2 + 2x + 1}{x^2 + 4x + 3} \), it's important to:

• Factorize both the numerator and the denominator completely.
• Identify and cancel out all common factors.

In the given example, both the numerator and the denominator were factorized. The common factor \( (x+1) \) is present in both the numerator and the denominator, allowing it to be canceled, simplifying the expression to \( \frac{x+1}{x+3} \). The key is understanding that only common factors multiplied (not added or subtracted) within the terms can be canceled.

Quadratic Expressions

Quadratic expressions are algebraic expressions where the highest exponent of the variable is 2. They come in the standard form \( ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants. Quadratics can often be factored into products of linear expressions or solved using other methods like the quadratic formula.In the expression \( x^2 + 4x + 3 \), it fits the quadratic form. Factoring such an expression involves finding two numbers that multiply to \( c \) (here 3) and add to \( b \) (here 4). For \( x^2 + 4x + 3 \), these numbers are 1 and 3, making the factorization \( (x+1)(x+3) \).These expressions frequently appear in various algebra problems, requiring mastery of factoring and simplification techniques to solve equations, simplify complex expressions, or interpret their graphical representations.