Problem 31 - Beginning and Intermediate Algebra

Question

Factor completely. $$12 c^{3}+3 c^{2}+27 c$$

Step-by-Step Solution

Answer

The completely factored expression is: \(3(4c^3 + c^2 + 9c)\).

Step 1: Find the Greatest Common Divisor (GCD) of the coefficients

To find the GCD of the coefficients (12, 3, and 27), list their factors and find the largest factor they have in common. The factors of 12 are: 1, 2, 3, 4, 6, and 12 The factors of 3 are: 1, and 3 The factors of 27 are: 1, 3, 9, and 27 The largest (greatest) factor they all have in common is 3.

Step 2: Factor out the GCD

Factor out the GCD found in the previous step (3) from each term in the given expression: \(3(4c^3 + c^2 + 9c)\)

Step 3: Check for further factoring of the expression in parentheses

Observe the expression in parentheses: \(4c^3 + c^2 + 9c\). There are no further common factors among these terms, and it cannot be factored further.

Step 4: Final Answer

The completely factored expression is: \(3(4c^3 + c^2 + 9c)\).

Key Concepts

Greatest Common Divisor

When factoring polynomials, one of the first steps is to identify and factor out the greatest common divisor (GCD) of the terms. The GCD is the largest factor that divides each term of a polynomial without leaving a remainder. By determining this factor, we simplify the polynomial expression.

To find the GCD of the coefficients of a polynomial, list the factors of each coefficient.

• For 12, the factors are 1, 2, 3, 4, 6, 12.
• For 3, the factors are 1, 3.
• For 27, the factors are 1, 3, 9, 27.

The largest factor common to 12, 3, and 27 is 3. Hence, 3 is the GCD for this polynomial.

This step is crucial as it reduces the polynomial, making it easier to work with and further factor if possible.

Polynomial Expressions

Polynomial expressions are mathematical expressions made up of variables and coefficients, combined using addition, subtraction, and multiplication. A polynomial can have multiple terms, each consisting of a coefficient and a variable raised to a power.

In the exercise, the polynomial expression is given as: \(12c^3 + 3c^2 + 27c\). Each term in this polynomial consists of a coefficient (like 12, 3, or 27) and a variable part (like \(c^3\), \(c^2\), or \(c\)).

Polynomials are often written in standard form, which means the terms are arranged from highest to lowest degree of the variable. Understanding the structure of polynomial expressions is important, as it helps to simplify, evaluate, and manipulate these mathematical expressions effectively.

Factoring Techniques

Factoring polynomials involves breaking them down into simpler 'pieces' or factors that, when multiplied together, give you the original polynomial. This process begins with finding the greatest common divisor but may extend to more complex techniques.

### Methods of Factoring:

• **Factoring out the GCD:** This is the simplest form, where you remove the largest common factor of all terms.
• **Grouping:** This method involves rearranging and grouping terms to find common factors.
• **Quadratic trinomials:** Recognizing patterns can allow further factoring, especially with expressions resembling quadratics.

For the problem given, the expression after factoring out the GCD looks like this: \(3(4c^3 + c^2 + 9c)\). Here, no further simplification was possible using advanced techniques, as the remaining polynomial is already in its simplest form.

Mastering these techniques allows solving a wide range of polynomial equations efficiently by reducing them into basic, solvable units.