 # Different Types of Algebraic Expressions – Definitions, Formula, Examples

Algebraic expressions are the expressions that have variables, constants, and arithmetic operators. Mainly, we have three different types of algebraic expressions. Get to know more about those algebraic expression types in the following sections. You can check what is meant by an algebraic expression, definitions of various algebraic expressions, examples, and worked out problems in this article.

## Algebraic Expression Definition

An algebraic expression is a mathematical term that contains variables, constants along with mathematical operators like addition, subtraction, division, and multiplication. The example of an algebraic expression is 5x + 20y – 9. The different parts of an algebraic expression are variable, coefficient, operator, and constant. The definitions of these different parts of an algebraic expression are:

• Constant: It is a term whose value remains unchanged throughout the expression.
• Variable: It is an alphabetic letter whose value is unknown. It can take any value based on the situation.
• Coefficient: It is a numerical value added before the variable to modify the variable value.
• Operator: Mathematical operators are used in algebraic expressions to perform some math calculations on two or more expressions.

### Types of Algebraic Expressions

The algebraic expressions are further divided into 5 different types. Let us discuss each of these types in the following sections.

1. Monomial Algebraic Expression

2. Polynomial Algebraic Expression

3. Binomial Algebraic Expression

4. Trinomial Algebraic Expression

5. Multinomial Algebraic Expression

### Monomial Algebraic Expression

An algebraic expression that has only one non-zero term is known as the monomial.

Examples of monomials:

7a³b² is a monomial in two variables a, b

$$\frac { 2ax }{ 3y }$$ is a monomial in three variables a, x and y.

x² is a monomial in one variable x.

2y is a monomial in one variable y.

### Polynomial Algebraic Expression

An algebraic expression that has one, two, or more terms is known as the polynomial.

Examples of Polynomials:

3x + 4y is a polynomial in two variables x, y

4x² – 3xy + 6y² + 80 is a polynomial in two variables x, y

m + 5mn – 7mn² + nm² + 9 is a polynomial in two variables m, n

a³b + 4b²c + 6ab + 2ca + 5bc is a polynomial in three variables a, b, c

### Binomial Algebraic Expression

An algebraic expression that has two non-zero terms is known as the binomial.

Examples of binomials:

5x + 6y³ is a binomial in two variables x, y

a + b is a binomial in two variables a, b

p – q² is a binomial in two variables p, q

m²n + 6 is a binomial in two variables m, n

### Trinomial Algebraic Expression

An expression that has three non-zero terms is known as trinomial.

Examples of Trinomial:

p + q + r is a trinomial in three variables p, q, r

$$\frac { x² }{ 3 }$$ + ay – 6bz is a trinomial in three variables x, y, z

xy + x + 2y2 is a trinomial in two variables x and y.

### Multinomial Algebraic Expression

An algebraic expression that has two or more than three terms is known as the multinomial.

Examples of Multinomial:

w + x – y + 2z is a multinomial in four variables w, x, y, z.

a + ab + b + bc + cd is a multinomial of five terms in four variables a, b, c, and d.

5x⁸ + 3x⁷ + 2x⁶ + 5x⁵ – 2x⁴ – x³ + 7x² – x is a multinomial of eight terms in one variable x.

### Formulas

The general algebraic formulas we use to solve the expressions or equations are:

• (a + b)² = a² + 2ab + b²
• (a – b)² = a² – 2ab + b²
• a2 – b² = (a – b)(a + b)
• (a + b)³ = a³ + b³ + 3ab(a + b)
• (a – b)³ = a³ – b³ – 3ab(a – b)
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)

### Algebraic Expressions Questions

Question 1:

Add algebraic expressions 3x + 5y – 6z and x – 4y + 2z.

Solution:

The given algebraic expressions are 3x + 5y – 6z and x – 4y + 2z

While adding two or more algebraic expressions, add the like terms together.

3x + 5y – 6z + x – 4y + 2z = (3x + x) + (5y – 4y) + (2z – 6z)

= 4x + y – 4z

Therefore, the sum is 4x + y – 4z

Question 2:

Subtract the algebraic expressions 3x² – 6x – 4 from x + 5 – 2x²

Solution:

The given algebraic expressions are 3x² – 6x – 4, x + 5 – 2x²

While subtracting two or more algebraic expressions, perform the operation only between the like terms.

x + 5 – 2x² – (3x² – 6x – 4) = x + 5 – 2x² – 3x² + 6x + 4

= (x + 6x) + 5 + 4 – (2x² + 3x²)

= 7x + 9 – 5x²

Therefore, x + 5 – 2x² – (3x² – 6x – 4) = 7x + 9 – 5x²

Question 3:

Simplify the algebraic expression by combining the like terms

4(2x+1) – 3x – 2

Solution:

The given algebraic expression is 4(2x+1) – 3x – 2

Remove the braces

4(2x+1) – 3x – 2 = 8x + 4 – 3x – 2

= (8x – 3x) + 4 – 2

= 5x + 2

Therefore, 4(2x+1) – 3x – 2 = 5x + 2

Question 4:

Reduce the algebraic expression to its lowest term

$$\frac { (x² – y²)}{ (x + y) }$$

Solution:

The given algebraic expression is $$\frac { (x² – y²)}{ (x + y) }$$

We see that the numerator and denominator of the given algebraic fraction is polynomial, which can be factorized.

(x² – y²) = (x + y)(x – y)

$$\frac { (x² – y²)}{ (x + y) }$$ = $$\frac { (x + y)(x – y)}{ (x + y) }$$

Cancel te like term (x + y)

= (x – y)

Therefore, $$\frac { (x² – y²)}{ (x + y) }$$ = x – y.

### FAQs on Types of Algebraic Expressions

1. What are the types of algebraic expressions?

The five different types of algebraic expressions are monomial, binomial, trinomial, multinomial, and polynomial.

2. What are the rules for algebraic expressions?

The basic rules are to combine the like terms, constants for addition or subtraction. Remove any grouping symbols like paranthesis, brackets by multiplying factors. Use the exponential rule to remove grouping.

3. Write examples of algebraic expressions?

The examples of algebraic expression is x + 2y + 1, x² + 5xy + y³ + 9, x⁴y + 4x³y² + 16xy.

4. How to derive algebraic expressions?

An algebraic expression is a combination of constants, variables and algebraic operations (+, -, ×, ÷). We can derive the algebraic expression for a given situation or condition by using these combinations.

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