Question

Completely factor the expression.\(3 x^{2}-48\)

Short answer

The complete factorization of \(3x^{2} - 48\) is \(3(x - 4)(x + 4)\)

Step-by-step solution

Identify common factors

The given equation is \(3x^{2} - 48\). The first step is to look for any common factors in the terms. In this case, the common factor is 3. Thus, the entire expression can be divided by 3.

Divide the expression by the common factor

Dividing the entire expression by 3 will result in \(x^{2} - 16\).

Factor the remaining expression

The expression \(x^{2} - 16\) is a difference of squares and it can be factored as \((x - 4)(x + 4)\).

Final factorization

Therefore, the completed factorization of the given expression \(3x^{2} - 48\) is \(3(x - 4)(x + 4)\).

Key concepts

Common Factors in Algebra

In algebra, finding common factors simplifies expressions and makes other operations, like factoring quadratic expressions, more approachable. When you are asked to factor an expression such as \(3x^2 - 48\), the first step is to identify if there are any common factors in all terms. In this case, both terms are divisible by 3, making it the common factor.

After identifying the common factor, it's much like breaking down a number into smaller, more manageable pieces. Divide each term by the common factor to simplify the expression. For \(3x^2 - 48\), when you divide by the common factor 3, you get \(x^2 - 16\), a much cleaner and more factorable quadratic expression. This initial step is crucial because it often reveals hidden patterns that lead to complete factorization of the expression.

Difference of Squares

The difference of squares is a unique pattern that shows up often in algebraic factorization. It is a key concept to recognize as it can be factored easily. A difference of squares takes the form of \(a^2 - b^2\), where \(a\) and \(b\) are any expressions. This pattern can always be factored into \((a - b)(a + b)\).

Looking at our simplified expression \(x^2 - 16\), from the common factor step, we can see that it fits this pattern, where \(a = x\) and \(b = 4\), since \(4^2 = 16\). Utilizing the difference of squares rule lets us break this further into the product \((x - 4)(x + 4)\). This crucial insight simplifies what can first appear as a complex or unsolvable problem into a much simpler one, emphasizing the importance of pattern recognition in algebra.

Algebraic Factorization

Algebraic factorization is the process of breaking down algebraic expressions into their simplest building blocks or 'factors'. It's an essential skill that can help in solving equations, simplifying expressions, and understanding mathematical relationships.

When factoring expressions like \(3x^2 - 48\), after removing common factors and recognizing special patterns such as the difference of squares, combining these steps allows us to reach a complete factorization: \(3(x - 4)(x + 4)\). In exercising algebraic factorization, not only do we find the factors, but we also gain insight into the structure of algebraic expressions and solutions to quadratic equations.