Question

Completely factor the expression.$12x^3-48x$

Short answer

The completely factored expression is: 12x(x-2)(x+2)

Step-by-step solution

Identifying Common Factors

The first step in solving this problem is to identify the common factors of the terms in the expression. Looking at the expression $12x^3-48x$, it can be noted that 12 and $x$ are common factors.

Factoring Out the Common Factors

Factoring out the common factors means to rewrite the expression as a product of the common factors and the remaining terms. Here, by factoring out 12x, the expression can be written as $12x(x^2-4)$.

Factoring the Quadratic Expression

The expression inside the brackets, $x^2-4$, is a difference of squares, which can be factored as $(x-2)(x+2)$. Thus, the completely factored expression is given by $12x(x-2)(x+2)$.

Key concepts

Common Factors in Algebra

Understanding common factors in algebra is vital to simplifying expressions and solving equations efficiently. When we talk about common factors, we refer to numbers or variables that are shared among terms in an algebraic expression. To spot these, look for the highest number and any variables that each term has in common. For example, in the expression 12x^3 - 48x, both terms include a multiple of 12 and at least one instance of x.

Factoring these out simplifies the expression: we rewrite it as the product of the common factors and what’s left after dividing each term by those factors. In our case, 12x is factored out, transforming the original equation into 12x(x^2 - 4). This pivotal step makes subsequent factoring steps clearer and sets the stage for breaking down more complex structures within the algebraic expression.

Factoring Quadratic Expressions

Quadratic expressions, which are polynomials of degree two, often take the form ax^2 + bx + c. Factoring these expressions is a fundamental skill in algebra and can be achieved through various methods, including applying the difference of squares approach, utilizing the quadratic formula, or factoring by grouping. The goal is to break down the quadratic into a product of binomials.

In our example x^2 - 4, once we have factored out the common factor 12x, we are left with a quadratic expression that happens to be a perfect square difference. Recognizing these opportunities allows us to continue factoring efficiently. Still, it requires familiarity with patterns and practice with different factoring techniques to handle more complex quadratics that may not lend themselves to difference of squares or have coefficients other than 1 for x^2.

Difference of Squares

The difference of squares is a special and handy pattern in algebra that arises when a quadratic expression can be written as the subtraction of two perfect squares. Its standard form is a^2 - b^2, which can be factorially decomposed into (a - b)(a + b). This powerful tool in algebra simplifies expressions and solving equations.

For instance, a closer look at the quadratic portion of our example, x^2 - 4, shows that it fits the difference of squares pattern perfectly. The number 4 is a perfect square because it is equal to 2^2, and x^2 is itself a perfect square of x. Following the pattern, we can express x^2 - 4 as (x - 2)(x + 2). Identifying this pattern allows students to factor quickly without resorting to other more time-consuming methods, such as the quadratic formula or completing the square.