1. Reviewing Algebraic Terms

Learning outcomes
  • I can identify variables, constants, coefficients, and terms.
  • I can distinguish between expressions and equations.
  • I can identify like and unlike terms.
  • I can describe the degree of a term.
  • I can use correct algebraic vocabulary.

The Language of Algebra

Algebra uses letters, numbers, and mathematical operations to represent relationships.

Consider the algebraic expression:

Although this expression is short, it contains several important algebraic ideas.

  • x is a variable.
  • 5 is a coefficient.
  • 3 is a constant.
  • 5x and 3 are terms.

Understanding this vocabulary is important because we will use it when solving equations, simplifying expressions, graphing functions, and working with systems of equations.

Variables

A variable is a letter or symbol used to represent an unknown or changing value.

For example:

The variable is:

x​

Variables can use many different letters:

x, y, a, b, m, n

For example:

contains two variables:

a and b​

Constants

A constant is a number that does not contain a variable.

Consider:

The constant is: 9​

In:

the constant is: 12​

A constant has a fixed value.

Coefficients

A coefficient is a numerical factor multiplying a variable or variable expression.

Consider: 7x

The coefficient of x is: 7​

In: −4y

the coefficient is: −4​

In: 3x2

the coefficient is: 3​

The exponent does not change the coefficient.

Hidden Coefficients

Sometimes a coefficient is not written.

Consider: x

This actually means: 1x

Therefore, the coefficient is: 1​

Similarly: −x

means: −1x

So the coefficient is: −1​

Terms

A term is a single number, a variable, or a product of numbers and variables.

Terms in an expression are separated by addition or subtraction.

Consider:

The terms are: 5x, 3y, −8​

Notice that the negative sign belongs to the final term.

We should identify the term as: −8

rather than simply 8.

Identifying the Parts of an Expression

Consider:

We can identify:

Variables x

Coefficients 4, −7

Constant 9

Terms 4x2 , −7x, 9

This expression contains three terms.

Expressions and Equations

An algebraic expression is a combination of numbers, variables, and mathematical operations.

For example:

This is an expression.

It does not contain an equals sign.

An equation states that two expressions are equal.

For example:

This is an equation because it contains an equals sign.

Expression vs Equation

Expression Equation
No equals sign Contains an equals sign
Can be simplified or evaluated.   Can be solved

A useful rule is:

Equation ⇒ equals sign​

What Does It Mean to Evaluate an Expression?

To evaluate an expression means to calculate its value when the variable values are known.

Consider:

If:

then:

Therefore: 3x + 5 = 17 when x = 4​

Expressions are usually evaluated or simplified, while equations are usually solved.

Like Terms

Like terms contain exactly the same variable parts with the same exponents.

Consider:

Both terms contain x.

Therefore, they are like terms.

We can combine them:

More Examples of Like Terms

are like terms.

3x2 and 8x2

are like terms.

are like terms.

Constants are also like terms:

4 and 11

Unlike Terms

Unlike terms have different variable parts or different exponents.

For example: 3x and 3y

are unlike terms because they contain different variables.

Similarly: 5x and 5x2

are unlike terms because the exponent on x is different.

And: 2xy and 2x

are unlike terms because their variable parts are different.

Why Can We Only Combine Like Terms?

Consider:

If x represents apples, this is similar to:

which gives:

7 apples

Therefore:

But:

would be like:

We cannot combine them into 7 of the same thing.

Therefore: 3x + 4y​

cannot be simplified by combining the terms.

Combining Like Terms

Consider:

First identify the like terms.

Variable terms:

Constants:

Now combine them:

Therefore:

4x + 3 + 7x − 5 = 11x −2​

Example with Two Variables

Simplify:

Group the x-terms:

Group the y-terms:

Therefore:

3x + 9y​

We cannot combine 3x and 9y because they are unlike terms.

Exponents

An exponent tells us how many times a quantity is multiplied by itself.

For example:

x3

means:

The exponent is:

3​

Similarly:

If no exponent is written:

The exponent is understood to be 1.

Degree of a Term

The degree of a term is the sum of the exponents on its variables.

Consider:

5x3

The exponent on x is 3.

Therefore:

Degree = 3​

The coefficient 5 does not affect the degree.

More Examples

Example 1

7x

Remember:

Therefore:

Degree = 1​

Example 2

4x5

Therefore:

Degree=5​

Example 3

6x2y

Remember:

Add the exponents:

Therefore:

Degree = 3​

Example 4

8a2b3

Add the exponents:

Therefore:

Degree = 5​

What About Constants?

A nonzero constant such as:

7

contains no variables.

Its degree is:

0​

So:

Term  Degree 
8 0
3x 1
5x2 2
−2x3 3
4x2y 3
6a2b3 5

Don't Confuse Coefficient and Exponent

Consider:

5x3

The coefficient is:

5​

The exponent is:

3​

The degree is:

3​

These numbers have completely different meanings.

The coefficient tells us the numerical factor multiplying the variable.

The exponent tells us the power of the variable.

Putting It All Together

Consider:

Variable

x​

Terms

6x2, −4x, 11​

Coefficients

6, −4​

Constant

11​

Degrees of the Terms

Example with Two Variables

Consider:

Variables

x, y​

Terms

3x2y, −7xy, 4y, −9​

Coefficients

3, −7 , 4​

Constant

−9​

Degrees

For:

3x2y

Degree = 3.

For:

−7xy

Degree = 2.

For:

4y

Degree = 1.

For:

−9

Degree = 0.

Common Algebra Mistakes

Mistake 1: Calling every number a coefficient

In:

5 is a coefficient, but 8 is a constant.

Mistake 2: Combining unlike terms

Incorrect:

Correct:

3x + 4y​

The terms cannot be combined.

Mistake 3: Ignoring negative signs

In:

the second term is:

−3y​

not simply 3y.

Mistake 4: Confusing coefficients with exponents

In:

7x4

Algebraic Vocabulary in Context

Consider:

A mathematically precise description would be:

This is an equation containing two variables, x and y. The expression on the left contains two terms, 2x and 3y. Their coefficients are 2 and 3. The number 12 is a constant.

Using correct mathematical vocabulary allows us to communicate mathematical ideas clearly.

Did You Know?

The word algebra comes from the Arabic word al-jabr, which appeared in the title of a mathematical work written by the Persian mathematician al-Khwarizmi around the ninth century.

Modern algebra has developed far beyond solving simple equations. It is now an essential language used throughout mathematics, physics, engineering, economics, computing, and many other fields.

Key Terms

  • Variable – a letter or symbol representing an unknown or changing value.
  • Constant – a fixed numerical value without a variable.
  • Coefficient – a numerical factor multiplying a variable or variable expression.
  • Term – a number, variable, or product of numbers and variables.
  • Expression – a combination of numbers, variables, and mathematical operations without an equals sign.
  • Equation – a mathematical statement showing that two expressions are equal.
  • Like terms – terms containing the same variables raised to the same powers.
  • Unlike terms – terms with different variable parts or exponents.
  • Exponent – a number indicating the power to which a quantity is raised.
  • Degree of a term – the sum of the exponents of the variables in a term.
  • Evaluate – calculate the value of an expression for specified variable values.
  • Simplify – rewrite an expression in an equivalent but simpler form.

Key Takeaways

  • A variable represents an unknown or changing quantity.
  • A constant is a fixed number without a variable.
  • A coefficient is the numerical factor multiplying a variable.
  • An algebraic expression is made from individual terms.
  • An expression does not contain an equals sign, while an equation does.
  • Like terms have identical variable parts and can be combined.
  • Unlike terms cannot be combined through addition or subtraction.
  • The degree of a term is found by adding the exponents of its variables.
  • A nonzero constant has degree 0.
  • Correct algebraic vocabulary helps us describe and solve mathematical problems accurately.