Problem 77 - Algebra and Trigonometry Real Mathematics, Real People

Question

Factor the polynomial completely. \(x^{3}-16 x\)

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(x^{3}-16 x\) is \(x(x-4)(x+4)\).

Step 1: Identify a Common Factor

First, observe the polynomial term by term. In this case, there exists a common factor, \(x\), in both terms, which is the first step of factoring any expression. Therefore, factor out \(x\) from the given cubic polynomial.

Step 2: Factor out the Common Factor

After identifying the common factor \(x\), the next step is to factor out \(x\) from \(x^{3}-16x\). By doing this, the equation becomes \(x(x^{2}-16)\).

Step 3: Use the Difference of Squares Formula

The next step involves recognizing that the polynomial is now a difference of squares, which can be factored using the formula \(a^{2}-b^{2}=(a-b)(a+b)\). Here, \(a=\ x\) and \(b=4\). By substituting these values in the formula, the factored form of \(x^{2}-16\) would be \((x-4)(x+4)\).

Step 4: Write the Final Factored Form

By substituting the factored form of \(x^{2}-16\) back to the equation \(x(x^{2}-16)\), the final factored form of the polynomial \(x^{3}-16x\) would be \(x(x-4)(x+4)\). This represents the polynomial in its completely factored form.

Key Concepts

Difference of Squares

The difference of squares is a special algebraic pattern and a key concept in polynomial factoring. It applies to expressions of the form \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\). This pattern is useful because it allows for quick factoring of certain quadratic expressions.

In the case of the polynomial given, \(x^2 - 16\), we can recognize it as a difference of squares where \(a = x\) and \(b = 4\). Factoring it using the difference of squares formula results in \((x-4)(x+4)\).

Using the difference of squares simplifies expressions and is an essential tool for algebra students to master. It is often a step in many factoring problems, making it a very powerful technique.

Common Factor

Identifying a common factor is often the first step in factoring polynomials. A common factor is a term that can be divided out from each term of the polynomial. Recognizing this helps simplify the equation significantly.

In our polynomial \(x^3 - 16x\), the common factor is \(x\). By factoring out \(x\), the expression becomes manageable, bringing us to \(x(x^2 - 16)\).

Factoring out the greatest common factor (GCF) is important because it makes other factoring steps more apparent and accessible. Once the common factor is removed, other patterns, like the difference of squares, often become clearer.

Cubic Polynomial

A cubic polynomial is any polynomial with the highest degree of three. This degree signifies that the variable will be raised to the third power. Factoring cubic polynomials often begins with finding a common factor, followed by applying other factoring methods.

For the polynomial \(x^3 - 16x\), initially, it appears complex. However, by targeting the common factor \(x\), it can be reduced to quadratic form \(x^2 - 16\), which then allows for further factoring using techniques like the difference of squares.

Understanding the nature of cubic polynomials helps in breaking them down through systematic steps to achieve a completely factored form.

Polynomial Factoring Steps

Polynomial factoring involves breaking down a polynomial into simpler terms or expressions. These simpler terms multiply to give the original polynomial. Following a structured approach ensures accuracy and clarity in solving these problems.

The common steps include:

• Identifying and factoring out the greatest common factor.
• Simplifying the polynomial to reveal other factoring opportunities, like the difference of squares.
• Applying specific formulas or methods based on the polynomial’s characteristics, such as \(a^2 - b^2 = (a-b)(a+b)\).
• Combining all factored parts to express the original polynomial in a completely factored form.

In our problem, by following these steps, the polynomial \(x^3 - 16x\) is fully factored into \(x(x-4)(x+4)\). This step-by-step approach is crucial for mastering polynomial factoring effectively.