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Chapter 2: Problem 16

Write the expression in a simpler form, if possible. . \(3 a+5 a b+2 b\)

Short Answer

Expert verified
Answer: The simplest form of the expression is \(3a+5ab+2b\), since there are no like terms to combine.

Step by step solution

01

Identify like terms

In the given expression, \(3a+5ab+2b\), we observe that there are no like terms since each term has different variables or products of variables.
02

Final result

After observing that there are no like terms in the expression, we conclude that the given expression, \(3a+5ab+2b\), is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Like Terms
When simplifying algebraic expressions, identifying 'like terms' is a key step. Like terms are terms that have the exact same variables raised to the same powers. For example, in the expression \(3a + 5ab + 2b\), like terms would be any terms that have exactly the same variable component. However, in our example:
  • \(3a\) has a single variable \(a\).
  • \(5ab\) includes variables \(a\) and \(b\), making it distinct from \(3a\).
  • \(2b\) contains only \(b\).
Because each term consists of different variables or combinations thereof in our given example, there are no like terms that can be combined. This is why the expression \(3a + 5ab + 2b\) remains in its simplest form. Understanding like terms helps us know when we can combine terms to simplify expressions.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operations (such as addition, subtraction, multiplication, or division). They can range from simple expressions like \(3x + 2\) to more complex ones involving multiple variables and powers. In our example, \(3a + 5ab + 2b\), we have an algebraic expression that combines both 'monomials' and 'binomials.'
The structure of an algebraic expression can be broken down into:
  • **Constants**: Numbers on their own (such as 2 in another context).
  • **Variables**: Symbols that stand in for unknown values (here, \(a\) and \(b\)).
  • **Coefficients**: Numbers that multiply the variables (like 3 in \(3a\)).
Understanding how these elements interact within an expression allows us to manipulate and simplify them to solve equations or model real-world situations.
Variables and Coefficients
In an algebraic expression, variables and coefficients are fundamental components. A variable is a symbol, usually a letter, that represents a number. In the expression \(3a + 5ab + 2b\):
  • \(a\) and \(b\) are variables representing unknown or changeable numbers.

Coefficients, on the other hand, are the numbers placed before variables to indicate multiplication. They tell us how many times a number should be counted with the variable it multiplies.
  • In \(3a\), the number 3 is the coefficient of \(a\).
  • In \(5ab\), 5 is the coefficient of the compound term \(ab\).
Understanding the role of variables and coefficients helps in simplifying expressions, solving equations, and understanding algebraic relationships. When these elements are combined, they form the terms of an expression, which are the building blocks of algebra.

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Most popular questions from this chapter

Write each expression as a sum, difference, or product of two or more algebraic fractions. There is more than one correct answer. Assume all variables are positive. $$ \frac{c}{a b} $$

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