Problem 33 - 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. $$3 x^{3}+27 x$$

Step-by-Step Solution

Answer

The polynomial factored completely is \(3x(x^2 + 9)\).

Step 1: Identify Common Monomial Factor

Look for the greatest common factor (GCF) in the polynomial. The terms are \(3x^{3}\) and \(27x\). Both terms have a common factor of \(3x\).

Step 2: Factor Out the GCF

Divide each term of the polynomial by the GCF \(3x\) and factor it out: \[ 3x(x^2 + 9) \]

Step 3: Check for Further Factoring

Examine the quadratic \(x^2 + 9\). This expression is a sum of squares, which cannot be factored over the integers. There are no further integer factors beyond the GCF already factored out.

Key Concepts

Greatest Common Factor

When you see a polynomial like the one in the exercise, the first step to factor it is to look for the greatest common factor (GCF) of all the terms. The GCF is the largest factor that divides all the terms evenly.

• Take a close look at the polynomial's terms, such as \(3x^{3}\) and \(27x\).
• Here, both terms share a common factor of \(3x\).

Once identified, you factor this GCF out of the polynomial. This means dividing each term by the GCF and writing the polynomial as the product of the GCF and the resulting expression. This simplifies the polynomial and is a crucial step in the process of factoring completely.

Sum of Squares

After you factor out the greatest common factor, you may encounter expressions like \(x^2 + 9\). These types of expressions are known as the "sum of squares." Unlike the "difference of squares," which can be factored over the integers, the sum of squares is not factorable into real numbers or integers.
You might remember that the sum of squares formula, \(a^2 + b^2\), doesn't have a straightforward factorization with integers. So, in this case:

• \(x^2 + 9\) remains unchanged when considering integer factorization.
• There are no two integers that you can multiply to make -9 (as required to factor it like a difference of squares), so this is as far as integer factorization goes.

Understanding this concept helps you see why certain terms may remain in their original form when factoring over integers.

Integer Factorization

Integer factorization involves breaking down a polynomial until all factors are expressed using integers. It's a vital skill in simplifying polynomial expressions. Let's see how it works in the context of our example.

• First, begin with factoring out common factors as we did with the greatest common factor.
• After removing the GCF, examine any remaining expressions to see if they can be factored further using integers.

In our case, we factored out \( 3x \) to get \( 3x(x^2 + 9) \). We then checked \( x^2 + 9 \) for possible further factoring. However, since it's a sum of squares, it doesn't factor nicely into integers.
The importance of integer factorization lies in simplifying expressions—making them easier to manage and solve in equations. Remember, a fully factored polynomial has all its parts expressed with integers whenever possible, except in cases like the sum of squares.