Problem 23 - Elementary Algebra

Question

For the expressions in the following problems, write the number of terms that appear and then list the terms. $$ 5 x^{2}+6 x-2 $$

Step-by-Step Solution

Answer

Answer: There are 3 terms in the given polynomial expression. They are \(5 x^{2}\), \(6 x\), and \(-2\).

Step 1: Identify the terms in the expression

First, look at the given expression: $$ 5 x^{2}+6 x-2 $$ This expression has three terms: 1. \(5 x^{2}\) 2. \(6 x\) 3. \(-2\)

Step 2: Count the terms and list them

In the expression, we have identified three terms. Let's list them in the answer: 1. \(5 x^{2}\) 2. \(6 x\) 3. \(-2\) Thus, the number of terms in the given polynomial expression is 3 and the terms are \(5 x^{2}\), \(6 x\), and \(-2\).

Key Concepts

Terms in Algebra

In algebra, a **term** is a single mathematical expression. It could be a number (also called a constant), a variable (like \(x\) or \(y\)), or a combination of numbers and variables multiplied together. Terms are separated by plus or minus signs.
For example, in the expression \(5x^2 + 6x - 2\), each part separated by a plus or minus sign is a term:

• \(5x^2\) is a term because it is a single expression with one variable, \(x\), raised to a power and multiplied by a coefficient 5.
• \(6x\) is another term. It represents a variable \(x\) multiplied by the coefficient 6.
• \(-2\) is a term too, called a constant term, because it stands alone without a variable attached to it.

The versatility of terms allows you to construct complex algebraic expressions that model real-world situations or solve problems efficiently.

Counting Terms

Counting terms is simple but essential when working with algebraic expressions. To find out how many terms are in an expression, look at the entirety of the expression and count each unique segment that stands alone:

• Identify the parts of the expression separated by plus or minus signs. Each segment you see is a term.
• Remember, even if a term does not explicitly show a coefficient, every variable has an implicit coefficient of 1.
• Subtracting affects the sign of a term but not its status as a separate term. So, \( -2 \) is still only one term.

In the expression \(5x^2 + 6x - 2\), you can easily count three terms: \(5x^2\), \(6x\), and \(-2\). Take each section one by one, and you will become an expert at recognizing terms in no time!

Algebraic Expressions

An algebraic expression is a mixture of constants, variables, and operations. It’s a way of writing a number that uses letters to represent variables. Expressions can be as simple as just one term (like \(x\) or 7) or very complex involving multiple operations.

Here are some key points about algebraic expressions:

• They do not have an "equals" sign. They represent a value or set of values.
• They can include one or more terms. For example, \(5x^2 + 6x - 2\) has three terms, making it a polynomial expression.
• They can represent real-world information, like calculating distances, profits, or physics problems.

Recognizing the structure of an algebraic expression allows you to manipulate it to solve equations, factorize expressions, or simplify them for practical use. Understanding this key algebraic concept opens the door to solving a wide range of mathematical problems.