Problem 30 - Algebra 1: Concepts and Skills

Question

Simplify the expression. If not possible, write already in simplest form. $$\frac{x^{2}+8 x+16}{3 x+12}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\frac{x^{2}+8 x+16}{3 x+12}\) is \(\frac{x+4}{3}\).

Step 1: Factorize the numerator

Factorizing quadratics, the numerator \(x^{2}+8 x+16\) can be written as \((x+4)^2\).

Step 2: Factorize the denominator

The denominator \(3 x+12\), similarly, can be rewritten as \(3(x+4)\).

Step 3: Simplify the expression

After factorization, the expression becomes \(\frac{(x+4)^2}{3(x+4)}\). Then, we cancel out the common factor \((x+4)\) from both numerator and denominator. This results in the expression \(\frac{x+4}{3}\).

Key Concepts

Quadratic Factorization

When dealing with quadratic expressions, factorization is a powerful tool to simplify them. A quadratic expression is typically of the form \(ax^2 + bx + c\). In our specific example, we have \(x^{2} + 8x + 16\), a perfectly formed trinomial. This kind of expression can be rewritten using two identical binomials, like \((x+4)^2\).
By recognizing the pattern, you can see that 8 is twice the number 4 (from \(x+4\)) multiplied by one of the \(x+4\) terms. Thus, you split the linear term (8x) as twice the product of 4 and x, and the constant term (16) as 4 squared, indicating a square pattern:

• The expression can be identified as a perfect square trinomial and factorized as such.
• This is a specific case where the quadratic follows \(a^2 + 2ab + b^2 = (a+b)^2\).

Recognizing this quickly makes it easier to factorize and simplify the expression promptly.

Polynomial Division

Polynomial division is akin to numerical long division but slightly different due to variables and exponents. The goal is often to simplify expressions by "dividing" out common polynomial factors.
In this exercise, once we have factorized the quadratic, we encounter a simpler polynomial expression instead. The numerator becomes \((x+4)^2\) and the denominator \(3(x+4)\).
These terms indicate we have a common polynomial factor of \(x+4\) that we divide out to streamline the expression. Polynomial division in this manner serves to simplify, reducing workloads in subsequent calculations. Once you identify shared factors, divide the whole expression by the common polynomial factor.

• By reducing them, the expression is minimized to only terms that influence the solution.
• The result is a simpler expression that's easier to handle in future calculations or to interpret directly from the problem.

Canceling Common Factors

When simplifying complex expressions, canceling common factors is a fundamental skill that helps arrive at the simplest form. To effectively cancel out common factors, the factorization step is crucial.
In the example, the factorized expression \(\frac{(x+4)^2}{3(x+4)}\) illustrates how a shared factor \((x+4)\) appears in both the numerator and the denominator.
Only common factors can be canceled, meaning they must be identical and appear once in both the numerator and the denominator. It is essential to ensure there are no zero denominators as this would invalidate the expression.

• Start by factoring both numerator and denominator to reveal all common terms.
• Cancel out the shared terms, which when simplified provides \(\frac{x+4}{3}\), an easier expression to work with.

Remember, canceling is permissible as long as the division by zero is safeguarded against, thus maintaining the integrity of the algebraic expression.