Problem 3 - Algebra 2

Question

Simplify each expression. \(\frac{x^{2}+6 x+9}{x+3}\)

Step-by-Step Solution

Answer

The expression simplifies to \(x+3\).

Step 1: Identify the expression's components

The given expression is \(\frac{x^{2}+6x+9}{x+3}\). The numerator is a quadratic expression, \(x^2 + 6x + 9\), and the denominator is a linear expression, \(x + 3\).

Step 2: Factor the numerator

Notice that the numerator \(x^2 + 6x + 9\) is a perfect square. It can be rewritten as \((x+3)^2\) because expanding \((x+3)(x+3)\) gives \(x^2 + 6x + 9\).

Step 3: Write the expression using the factored numerator

Substitute the factorized numerator into the expression: \(\frac{(x+3)^2}{x+3}\).

Step 4: Cancel common factors

Both the numerator \((x+3)^2\) and the denominator \(x+3\) have a common factor \(x+3\). Cancel out the \(x+3\) from the numerator and denominator to simplify the expression to \(x+3\).

Step 5: State the simplified expression

After canceling the common factor, the expression simplifies to \(x + 3\). This is the final simplified form.

Key Concepts

Factoring Quadratics

Understanding how to factor quadratics is crucial when simplifying algebraic expressions. Quadratics are polynomial expressions of degree two, typically in the form \( ax^2 + bx + c \). Factoring them involves breaking them down into simpler expressions that multiply to give the original quadratic.
Consider the quadratic expression \( x^2 + 6x + 9 \). To factor it, look for two numbers that multiply to the last term, 9, and add up to the middle coefficient, 6. Here, the numbers 3 and 3 work, making the factors \((x+3)(x+3)\).

• Factoring helps in both solving equations and simplifying expressions.
• Check your work by expanding the factors to ensure they match the original quadratic.
• Quadratics can sometimes be tricky, so practice identifying patterns.

Remember, not all quadratics are factorable by integers, but recognizing patterns can still help simplify or solve expressions.

Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form \( (a+b)^2 \) or \( (a-b)^2 \), which when expanded are \( a^2 + 2ab + b^2 \) and \( a^2 - 2ab + b^2 \) respectively.
In our example, \( x^2 + 6x + 9 \) can be factored into \( (x+3)^2 \). This is because:

• The square of \( x \) is \( x^2 \).
• Twice the product of \( x \) and 3 is \( 6x \).
• The square of 3 is 9.

Identifying perfect square trinomials helps simplify expressions quickly. When you see a trinomial:

• Check if the first and last terms are perfect squares.
• Verify if the middle term is twice the product of the square roots of the first and last terms.

These steps can greatly reduce complexity in algebraic manipulation.

Canceling Common Factors

Canceling common factors is an essential technique in simplifying fractions and rational expressions. It involves removing the same factor from both the numerator and the denominator, provided it's not zero, to simplify the expression.
Consider the expression \( \frac{(x+3)^2}{x+3} \). Both the numerator and the denominator have \( x+3 \) as a common factor. By canceling \( x+3 \) in both places, the expression simplifies to \( x+3 \).

• Only cancel common factors; ensure they are not equal to zero to avoid invalid simplifications.
• Cancelling helps in reducing complex expressions to simpler forms.
• After canceling, always verify that the simplified expression is valid.

This technique is useful in many areas of algebra, saving time and effort while solving or simplifying expressions.