Problem 37 - Algebra for College Students

Question

Factor each of the following polynomials completely. Indicate any that are not factorable using integers. Don't forget to look first for a common monomial factor. $$6 x-6 x^{3}$$

Step-by-Step Solution

Answer

The polynomial \(6x - 6x^3\) factors to \(6x(x - 1)(x + 1)\).

Step 1: Identify the Greatest Common Factor (GCF)

The polynomial given is \(6x - 6x^3\). First, identify the greatest common factor of the terms. Both terms share the factor of \(6x\).

Step 2: Factor Out the GCF

Factor out the greatest common factor, \(6x\), from the polynomial. This gives us: \[ 6x(x^2 - 1) \]

Step 3: Recognize and Factor the Difference of Squares

The expression \(x^2 - 1\) is a difference of squares, which can be factored further. The difference of squares formula is \(a^2 - b^2 = (a - b)(a + b)\). Apply this to get: \[ x^2 - 1 = (x - 1)(x + 1) \]

Step 4: Write the Complete Factorization

Substitute back the factored form of \(x^2 - 1\) into the expression from Step 2: \[ 6x(x - 1)(x + 1) \] This is the completely factored form of the polynomial.

Key Concepts

Greatest Common Factor

Before diving into polynomial factorization, it's important to first determine if there's a Greatest Common Factor (GCF) among all terms. The GCF is the largest factor that divides all the terms of the polynomial without a remainder. Finding this common factor simplifies the polynomial, making it easier to factor further.

• Look at the coefficients: For the polynomial \(6x - 6x^3\), both 6 is a common numerical factor.
• Look at the variable: The lowest power of 'x' common to both terms is \(x\).

By combining these, the GCF for \(6x - 6x^3\) is \(6x\). By factoring \(6x\) out, we reduce the expression to: \[ 6x(x^2 - 1) \]This step helps in simplifying the problem before applying other factoring techniques.

Difference of Squares

After factoring out the GCF from a polynomial, you might encounter a situation where the remaining expression fits the form of a Difference of Squares. The difference of squares is a special factoring case where you can express a term as \(a^2 - b^2\). This type of expression can be factored using the identity: \[ a^2 - b^2 = (a - b)(a + b) \]For example, in the expression \(x^2 - 1\) resulting from the previous step:

• Recognize that \(x^2\) is \(a^2\) with \(a = x\).
• Recognize that \(1\) is \(b^2\) with \(b = 1\).

So you can factor \(x^2 - 1\) as:\[ (x - 1)(x + 1) \]This important technique leverages the symmetrical nature of squares to simplify expressions further.

Factoring Techniques

Factoring is a process of breaking down polynomials into simpler terms or factors that when multiplied together give back the original polynomial. This process can sometimes seem challenging, but by following specific factoring techniques, the task becomes more manageable.

• **Identifying the GCF**: Always start by checking for the greatest common factor among the terms.
• **Difference of Squares**: Recognize patterns like \(a^2 - b^2\), which can be factored into \((a-b)(a+b)\).
• **Grouping**: This technique involves grouping terms to find a common factor in segments of the polynomial (not used here but an important tool to know).

The exercise \(6x - 6x^3\) highlights these techniques. By first pulling out the GCF and then recognizing a difference of squares, we saw how seemingly complex expressions could be broken down into straightforward multiplication of factors, yielding:\[ 6x(x - 1)(x + 1) \]Utilizing these basic principles can streamline the factoring process and enhance your ability to solve polynomial expressions efficiently.