91Ó°ÊÓ

The "difference of squares" is a special mathematical pattern used in polynomial factoring. When you have an expression of the form \(a^2 - b^2\), it breaks down neatly into \((a - b)(a + b)\). This pattern is useful because it simplifies the process of factoring polynomials that fit this structure.

To utilize this pattern, look for two perfect squares separated by a minus sign. In the exercise, after factoring out the common factor, the expression inside the brackets, \(x^2 - 16\), is a perfect example. The term \(x^2\) is \((x)^2\) and \(16\) is \((4)^2\).

Thus, applying the difference of squares rule, we factor \(x^2 - 16\) to \((x - 4)(x + 4)\). This technique simplifies complex polynomials and is a handy skill in algebra.

Identifying a common factor is usually the first step in factoring polynomials. A common factor is a term or number that divides exactly into each term in the expression. By pulling out this factor, the polynomial is simplified, making further factoring easier.

In the given problem, the polynomial is \(3x^2 - 48\). Both terms in this expression share a common factor, which is the number 3. Factoring out this 3 from each term, we rewrite the expression as \(3(x^2 - 16)\). This step is crucial as it prepares the expression for further simplification through other factoring techniques like the difference of squares.

Factoring techniques are a vital part of algebra, helping to break down polynomials into simpler components. These techniques transform complicated expressions into more manageable forms, often revealing solutions or roots more clearly.

In this exercise, two main techniques are applied. First, we identify and pull out a common factor from the polynomial, \(3x^2 - 48\), simplifying it to \(3(x^2 - 16)\). Then, recognizing the form inside the brackets as a difference of squares, \(x^2 - 16\) gets factored further to \((x - 4)(x + 4)\).

Finally, combine these insights to achieve the completely factored expression: \(3(x - 4)(x + 4)\). Learning these techniques builds a strong foundation in algebra, equipping you with tools to tackle a variety of polynomial expressions.