Problem 33 - Introductory Algebra for College Students

Question

Factor completely, or state that the polynomial is prime. $$3 x^{3}+27 x$$

Step-by-Step Solution

Answer

The completely factored form of the polynomial \(3x^{3}+27x\) is \(3x(x^{2}+9)\).

Step 1: Identify if the Polynomial is Prime

The given polynomial is \(3x^{3}+27x\). To identify if it is prime or not, look for any common factors. In this case, there is a common factor of \(3x\) in both terms.

Step 2: Factor out the Common Factor

By taking out the \(3x\) common factor, the polynomial becomes \(3x(x^{2}+9)\). This is the factored form of the polynomial.

Step 3: Check if the Factored Polynomial Can Be Further Factored

In the expression \(x^{2}+9\), no further factoring can be done as there are no two numbers that multiply to give 9 and add up to 0.

Key Concepts

Common Factors

When factoring polynomials, the first step is often to look for common factors. These are numbers or variables that are shared by all terms in the polynomial. For example, in the polynomial \(3x^{3}+27x\), both terms have a \(3\) and an \(x\) that can be factored out.

Factoring out the greatest common factor \(GCF\), which is \(3x\) in this case, simplifies the polynomial into a product of the \(GCF\) and another simpler polynomial. Hence, the original polynomial \(3x^{3}+27x\) is factored as \(3x(x^{2}+9)\). Always look for the \(GCF\) before attempting any other factoring techniques as it can make the rest of the process more manageable.

Prime Polynomials

A polynomial is considered prime if it cannot be factored into the product of two or more non-constant polynomials. This is analogous to prime numbers, which cannot be divided evenly by any number other than 1 and themselves.

After extracting common factors, if the remaining polynomial cannot be factored further using integer coefficients, it is considered a prime polynomial. In our case, after factoring out the common factor from \(3x^{3}+27x\), we get \(3x(x^{2}+9)\). The remaining factor, \(x^{2}+9\), cannot be factored further using real numbers since the roots of this quadratic equation are complex numbers. Therefore, \(x^{2}+9\) is prime with respect to the real numbers.

Polynomial Factorization

Polynomial factorization involves expressing a polynomial as the product of its factors, which may include constants, variables, powers of variables, and other polynomials. The process starts with looking for common factors and then applying various factoring techniques such as grouping, the difference of squares, sum/difference of cubes, or the use of the quadratic formula.

In the given exercise, once the common factor \(3x\) is factored out, we are left with the factor \(x^{2}+9\). This particular expression does not factor further with real coefficients. Hence, the fully factored form of the initial polynomial is \(3x(x^{2}+9)\). Understanding the steps to factor polynomials is essential in simplifying expressions and solving equations effectively.