Problem 51 - Elementary Algebra

Question

Write the number of terms that appear in the expression \(5 x^{2}+2 x-6+(a+b)\), and then list them.

Step-by-Step Solution

Answer

Answer: There are 3 individual terms in the expression.

Step 1: Combine constant terms

In the given expression, both -6 and (a+b) are constants. To simplify the expression, we can group the constants together: \(5x^2 + 2x + (-6 + a + b)\).

Step 2: Count the number of individual terms

Now that we have grouped the constants, we can see that there are 3 individual terms in the expression: 1. \(5x^2\) 2. \(2x\) 3. \(-6 + a + b\)

Step 3: List the terms

The individual terms in the given expression are: - \(5x^2\) - \(2x\) - \(-6 + a + b\) There are a total of 3 terms in the expression.

Key Concepts

Combining Like Terms

In algebra, combining like terms is a crucial step in simplifying expressions. Like terms are terms that have the same variable raised to the same power. For example, in the expression \(5x^2 + 2x + (-6 + a + b)\), terms like \(5x^2\) and \(2x\) are not like terms because they have different variable powers.
Meanwhile, constants such as \(-6\) and \(a + b\) can be grouped together. However, unless \(a\) and \(b\) have specified values, they remain as grouped constants and can't be condensed further.

• Why Combine? - Combining like terms makes an expression easier to work with and understand.
• Example: Consider \(3x + 4x\). They are like terms and can be combined into \(7x\).
• Non-Example: \(3x\) and \(3y\) cannot be combined because they have different variables.

Remember, the key is matching both the variable and its exponent, ensuring the terms are truly like terms.

Counting Terms in an Expression

Counting the number of terms in an expression is fundamental before any manipulation. An algebraic expression is typically composed of terms separated by plus or minus signs. These terms can include variables, constants, or a combination of both.
For the expression \(5x^2 + 2x + (-6 + a + b)\), let's break it down:

• First Term: \(5x^2\)
• Second Term: \(2x\)
• Third Term: \(-6 + a + b\)

Here, \(-6 + a + b\) is counted as one term because it is a group of constants that sit together as a single unit in this context.
When counting terms, focus on the separation by signs rather than grouping by parentheses.

Algebraic Expressions

Algebraic expressions comprise numbers, variables, and operations that come together to represent quantities. They do not have an equal sign and therefore do not make up an equation, which is something many students often confuse.
In the given problem, the expression \(5x^2 + 2x + (-6 + a + b)\) is a perfect example of an algebraic expression. It contains:

• Coefficients: Numbers multiplying the variables, like \(5\) in \(5x^2\) and \(2\) in \(2x\)
• Variables: Symbols representing quantities that can change, here \(x\), \(a\), and \(b\)
• Constants: Numbers on their own, namely \(-6\)

Understanding algebraic expressions is critical since they form the basis for algebra and higher-level math topics.
Compiling these components allows us to simplify, evaluate, and eventually solve related algebraic problems.