Problem 50 - Elementary Algebra

Question

For the following problems, simplify each of the algebraic expressions. $$ 6 x^{3}+12 x+5 $$

Step-by-Step Solution

Answer

Answer: The simplest form of the algebraic expression is \(6x^3 + 12x + 5\).

Step 1: Examine the given expression

Take a look at the given expression, \(6x^3 + 12x + 5\). There are three terms: one with an \(x^3\) term, one with an \(x\) term, and a constant term with no variable.

Step 2: Check for common factors

Now, we will check if there is a common factor across all terms. In this case, our terms are \(6x^3\), \(12x\), and \(5\). There is no common factor through all three terms.

Step 3: Check for like terms

Finally, let's check if there are any like terms to combine. Like terms are terms that have the same variable and exponent. In our case, \(6x^3\) has an \(x^3\) term, \(12x\) has an \(x\) term, and the constant term \(5\) has no variable. Therefore, there are no like terms to combine.

Step 4: Simplified expression

Since there are no common factors to extract and no like terms to combine, the given algebraic expression is already in its simplest form. Thus, the simplified expression is: $$ 6x^3 + 12x + 5 $$

Key Concepts

Algebraic Terms

Algebraic terms are the basic building blocks of algebraic expressions. Each term is made up of numbers and variables that are multiplied together. For example, in the term \(6x^3\), the number 6 is called the coefficient, while \(x^3\) is the variable part, indicating that \(x\) is raised to the power of 3.

Algebraic expressions can consist of one or more terms. In the problem provided, the expression \(6x^3 + 12x + 5\) has three individual terms:

• \(6x^3\)
• \(12x\)
• \(5\)

Each of these terms is separated by a plus or minus sign. Understanding the structure of algebraic terms helps in the process of simplifying expressions, identifying like terms, and finding common factors.

Common Factors

A common factor is a number or algebraic term that divides exactly into each term of an algebraic expression. Finding common factors is a crucial step in simplification because it helps to simplify the expression by dividing terms by their greatest common factor, if one exists.

In the expression \(6x^3 + 12x + 5\), we look to see if there is a number or algebraic term that evenly divides \(6x^3\), \(12x\), and \(5\).

• \(6x^3\) factors are 1, 2, 3, 6, \(x\), \(x^2\), \(x^3\)
• \(12x\) factors are 1, 2, 3, 4, 6, 12, \(x\)
• \(5\) factors are 1, 5

Here, no common factor exists across all terms, meaning the expression is already in its simplest form where common factors are concerned.

Like Terms

Like terms in an algebraic expression are terms that have exactly the same variable component—identical variables raised to the same power. This is a key concept when simplifying expressions since like terms can be combined together.

Consider the expression \(6x^3 + 12x + 5\). For the terms in this expression:

• \(6x^3\) is the only term with \(x^3\)
• \(12x\) is the only term with \(x\)
• \(5\) is a constant with no variable

Since none of these terms share both the same variable and the same exponent, we don't have any like terms to combine. When there are no like terms, the expression cannot be simplified further by combining terms.

Polynomials

Polynomials are a type of algebraic expression that consists of multiple terms, where each term is made up of a variable raised to a non-negative integer exponent and a coefficient.

The expression given, \(6x^3 + 12x + 5\), is a polynomial with three terms, also known as a trinomial because it has three separate terms.

Some key characteristics of polynomials include:

• The coefficients can be any real number.
• The exponents must be non-negative integers, such as 0, 1, 2, etc.
• The expression can be simplified or rearranged, but it does not contain any like terms in this case.

By understanding the structure of polynomials, we can better appreciate how each term contributes to the overall expression and why certain simplification processes can be applied.