Problem 27 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$2 x^{2}-18$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(2 x^{2}-18\) is \(2(x+3)(x-3)\).

Step 1: Identify Common Factor

Firstly, seek for a common factor in all terms. For \(2 x^{2}-18\), the common factor is 2.

Step 2: Factor out the Common Factor

The next step is to factor out the common factor. By factoring out 2 from \(2 x^{2}-18\), one gets \(2(x^{2}-9)\).

Step 3: Apply Difference of Squares Formula

Now, apply the difference of squares formula, \(a^{2}-b^{2}=(a+b)(a-b)\), to \(x^{2}-9\). This gives us \(2(x+3)(x-3)\).

Key Concepts

Common Factor

When you're factoring polynomials, spotting the common factor is often the first step. In algebra, a common factor is a number or expression that divides each term of the polynomial. For example, consider the expression 2x^2 - 18. It may not be immediately obvious, but the number 2 is a common factor of both terms since they are both divisible by 2.

Finding the common factor simplifies the factoring process because once identified, you can 'factor it out', or in other words, divide every term by that factor and pull it in front of the parenthesis. This is a critical first move before applying more complex factoring rules. For learners, think of this step as reducing the polynomial to a more manageable state before proceeding to other factoring techniques.

Difference of Squares

The next significant concept in factoring polynomials is the difference of squares. It refers to an expression in the form of a^2 - b^2, which can be factored into (a + b)(a - b). The rule is straightforward yet powerful and gets its name because you're dealing with two perfect squares separated by a subtraction sign.

For instance, once you've factored out a common factor from 2x^2 - 18 and are left with x^2 - 9, you are looking at a classic example of a difference of squares. Here, x^2 is the square of x, and 9 is the square of 3. By applying the difference of squares rule, you can quickly factor x^2 - 9 into (x + 3)(x - 3).

Factoring Out

The technique of factoring out involves taking the common factor previously identified and removing it from inside the polynomial by distributing it in front of a set of parentheses. This is an essential skill to master when dealing with algebraic expressions.

When you factor out the common factor from 2x^2 - 18, you divide each term by the common factor, which is 2, and write it next to each term within a parenthesis: 2(x^2 - 9). It's similar to breaking down the expression into a simpler form, where you've 'taken out' the factor of 2, reducing the inside of the parenthesis to something that can be further factored down using other methods.

Algebraic Expressions

Polynomials are a type of algebraic expression that contain variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. The polynomial 2x^2 - 18 is an example of a quadratic polynomial, which is an algebraic expression where the highest power of the variable, in this case, x, is 2.

Understanding how to handle algebraic expressions is fundamental to algebra. It involves recognizing patterns, like the difference of squares, and applying operations, like factoring out, to simplify or rearrange the expressions. Mastery of these skills leads to a better understanding of functions, equations, and eventually, calculus.