Problem 17 - Introductory Algebra for College Students

Question

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$5 x^{3}-20 x$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(5x^{3}-20x\) is \(5x(x+2)(x-2)\).

Step 1: Identify the Greatest Common Factor (GCF)

See if there is a GCF in \(5x^3 - 20x\), which could be factored out. Here, the GCF is \(5x\), since both terms in the polynomial are divisible by it.

Step 2: Factor out the GCF

Factor out the GCF from the polynomial. This will give: \(5x (x^2 - 4)\).

Step 3: Apply difference of squares

Notice that the expression inside the brackets is a difference of squares, \(x^2 - 4 = (x+2)(x-2)\). Substituting this back into the factored equation would provide the fully factored form of the polynomial.

Step 4: Replace difference of squares

The fully factored form of the initial expression would be \(5x(x+2)(x-2)\).

Key Concepts

Greatest Common Factor

The greatest common factor (GCF) is the largest factor that divides two or more numbers. When factoring a polynomial, finding the GCF is usually the first step. It involves identifying the highest possible term that can be extracted from every term in the polynomial.

In the polynomial given, \(5x^3 - 20x\), the terms are \(5x^3\) and \(20x\). Both terms share common factors of 5 and \(x\). When we extract these, we factor out the GCF, which is \(5x\).

After factoring out \(5x\), we are left with \(5x(x^2 - 4)\). This simplifies the polynomial and reduces its complexity, making the next steps in factoring easier. Always remember: the GCF is about simplification and setting the stage for more advanced factoring techniques.

Difference of Squares

A difference of squares is a specific type of polynomial that takes the form \(a^2 - b^2\). This can be factored into \((a+b)(a-b)\). It is an essential concept because it frequently appears in algebra. Recognizing this pattern allows us to simplify expressions rapidly.

For example, consider the expression \(x^2 - 4\) within the polynomial \(5x(x^2 - 4)\). Here, \(x^2\) is the square of \(x\), and \(4\) is the square of 2. Hence, \(x^2 - 4\) is a difference of squares. It can be rewritten as \((x+2)(x-2)\).

Recognizing and applying the difference of squares formula ensures we can factor complex expressions and find simpler, equivalent forms. This step is crucial for expressing a polynomial in its factorized form.

Factoring Techniques

Factoring is an essential algebra skill. It transforms complex expressions into simpler components, which are easier to handle. Various techniques are used based on the type and structure of the polynomial.

One common method involves identifying the greatest common factor, which reduces the degree of a polynomial and allows for further decomposition. Another technique is recognizing special products like the difference of squares, which provides patterns that can be factored easily.

For example, the expression \(5x(x^2 - 4)\) uses both techniques: extracting the \(5x\) as the GCF, and factoring \(x^2 - 4\) as a difference of squares. These techniques combined yield the fully factored form \(5x(x+2)(x-2)\).

Mastering these techniques is invaluable for simplifying polynomials, solving equations, and understanding the deeper properties of algebraic expressions.