Problem 30 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$2 x^{3}-8 x$$

Step-by-Step Solution

Answer

The factored form of the polynomial \(2x^{3}-8x\) is \(2x(x-2)(x+2)\).

Step 1: Identify the GCF

The greatest common factor between the terms \(2x^{3}\) and \(-8x\) is \(2x\). This is because \(2x\) is the largest factor that divides both terms evenly.

Step 2: Divide the Entire Polynomial by the GCF

Once the GCF has been identified, divide each term in the polynomial by this GCF. So \(2x^{3}\) divided by \(2x\) is \(x^{2}\) and \(-8x\) divided by \(2x\) is \(-4\).

Step 3: Write Down the Factored Form

The polynomial, after factoring, becomes the GCF times the result from the previous step, i.e. \(2x(x^{2}-4)\). However, note that \(x^{2}-4\) is a difference of squares, which can be further factored into \((x-2)(x+2)\). So a more complete factorization is \(2x(x-2)(x+2)\).

Key Concepts

Greatest Common Factor (GCF)

Factoring a polynomial often begins by identifying the Greatest Common Factor (GCF) among its terms. The GCF is the largest factor that divides each term evenly. In our example, the polynomial is \(2x^3 - 8x\). To determine the GCF:

• Look at the coefficients (the numbers directly in front of the variables). For \(2x^3\) and \(-8x\), the coefficients are 2 and -8. The largest number that divides both is 2.
• Examine the variable part. The variable in both terms is \(x\). Since \(x^3\) and \(x\) both have \(x\), we take the lowest power, which is \(x\).

The GCF for the polynomial is \(2x\), meaning each term can be divided by \(2x\) without leaving a remainder. Factoring out the GCF simplifies the polynomial and prepares it for further factorization.

Difference of Squares

The concept of a difference of squares comes into play when recognizing patterns within a polynomial. This happens when you have an expression like \(a^2 - b^2\). The difference of squares can be rewritten as the product of two binomials:

• The formula is \(a^2 - b^2 = (a - b)(a + b)\).

In our example, after factoring out the GCF, we found the expression \(x^2 - 4\). Notice \(x^2 - 4\) can be expressed as a difference of squares:

• Here, \(a^2\) is \(x^2\) and \(b^2\) is 4, which is also \(2^2\).

So, \(x^2 - 4\) becomes \((x - 2)(x + 2)\). This technique simplifies expressions and is crucial for achieving a complete factorization of a polynomial.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into simpler, easier-to-handle expressions, which multiply together to give the original polynomial. This process makes solving equations and simplifying expressions more manageable.
In our exercise, the polynomial \(2x^3 - 8x\) undergoes factorization as follows:

• Start by identifying the GCF, which is \(2x\).
• Divide the entire polynomial by \(2x\), leaving us with \(x^2 - 4\).
• Recognize \(x^2 - 4\) as a difference of squares and factor it into \((x - 2)(x + 2)\).

The completely factored form of the polynomial, now expressed as \(2x(x - 2)(x + 2)\), is made of linear factors. Each factor represents a simpler expression that builds up to the original polynomial when multiplied. Understanding and applying these steps ensures easier manipulation and comprehension of polynomials.