Algebraic expression is a powerful tool that can be used to represent and manipulate mathematical relationships. They are used in a wide range of fields, including mathematics, physics, engineering and computer science.

## What is an algebraic expression

An algebraic expression is a set of letters and numbers connected by mathematical operation signs.

- The letters are called variables because they can take different values.
- There are two categories of numbers: those that appear in front of variables are called coefficients and those that stand alone are called constants because they remain unchanged.

Variables are letters that represent unknown values. Constants are numbers that have a fixed value. Mathematical operations include addition, subtraction, multiplication, division and exponentiation.

Several algebraic expression examples:

**Linear Expression:**3x + 7

**Quadratic Expression :**2x^2 – 5x + 1

**Cubic Expression: **4x^3 + 2x^2 – 6x + 8

**Expression with Fractions: **(1/2)x – (3/4)y

**Expression with Exponents: **2x^3y^2 – 5xy^3

**Expression with Absolute Value : **|x – 3|

**Expression with Square Root : **√(4x + 9)

**Expression with Variables in Denominator : **1/(2x + 1)

**Expression with Trigonometric Functions : **sin(x) + cos(x)

**Expression with Logarithm : **log(2x + 1)

## How to write an algebraic expression

Writing an algebraic expression involves translating a mathematical phrase or statement into a concise mathematical form using variables and mathematical operations. Here’s a step-by-step guide on how to write an algebraic expression:

**Step 1: Identify the Variables**

Identify the quantities or variables involved in the problem or statement. These are typically represented by letters like x, y, a, b, etc.

**Step 2: Define What Operations to Perform**

Determine which mathematical operations are involved. Common operations include addition (+), subtraction (-), multiplication (*), division (/), exponentiation (^), square roots (√), absolute values (| |), and trigonometric functions (sin, cos, tan, etc.).

**Step 3: Translate Words to Operations**

Translate the words or phrases in the problem into mathematical operations. Here are some common phrases and their corresponding operations:

“Add” or “Sum of” ➡️ Addition (+)

“Subtract” or “Difference between” ➡️ Subtraction (-)

“Multiply” or “Product of” ➡️ Multiplication (*)

“Divide” or “Quotient of” ➡️ Division (/)

“Squared” or “Square of” ➡️ Exponentiation (^2)

“Cubed” or “Cube of” ➡️ Exponentiation (^3)

“Square root of” ➡️ Square root (√)

“Absolute value of” ➡️ Absolute value (| |)

**Step 4: Write the Expression**

Combine the variables and operations to write the algebraic expression. Use parentheses to clarify the order of operations if needed.

**Step 5: Simplify**

If possible, simplify the expression by performing any necessary calculations or reducing terms.

The key is to accurately represent the problem or relationship using variables and operations.

## Algebraic expression examples

5x (5 is the coefficient, and x is a variable)

3y + 12 (3 is the coefficient of y; y is the variable, and +12 is a constant)

7ab + 4b – 5 (7 and +4 are the respective coefficients of ab and b; a and b are variables, and -5 is a constant)

Note: There can also be numbers with exponents.

Example: 3 / 5x²y³ – 40 (3/5 is the coefficient of x²y³; x and y are variables; 2 and 3 are the respective exponents of x and y, and -40 is a constant)

➡️ Writing Textual Data as Algebraic Expressions:

We can express a situation (textual data) that we want to mathematize using an algebraic expression.

To do this, we need to assign a variable to each piece of data and then write the relationships between the data.

◼️ Example 1: Calculate the perimeter of a rectangular field that is 50 meters long and 30 meters wide.

1- We will assign a variable to each piece of data.

Let P be the perimeter of the rectangle; L its length, and l its width.

2- The relationship: The perimeter of a polygon is equal to the sum of its sides.

A rectangle has 4 sides: 2 lengths and 2 widths.

We write: P = 2L + 2l => P = 2(L + l)

By replacing L and l with their values, we will find the perimeter.

◼️ Example 2: If we subtract 19 from five times a number, we get 26.

We need to choose a variable to represent this number.

Let n be this number.

Five times n is: 5n

We subtract 19 from 5n (this is a subtraction):

5n – 19

We get 26 (this is the result):

5n – 19 = 26

◼️ Example 3: Translate the following sentence into an algebraic expression: “two times the square of the cube of triple a.”

For this type of sentence, we start from the end and work our way back to the beginning.

Triple a is: 3a

The cube of 3a is: (3a)³

The square of (3a)³ is: ((3a)³)²

Two times ((3a)³)² is: 2((3a)³)²

Attention: It is important to distinguish between the cube of triple a and the triple of the cube of a.

The cube of triple a is: 3a cubed. Therefore, parentheses are necessary: (3a)²

The triple of the cube of a is: a³ tripled: 3a³.

## Algebraic expression practice problems

**Problem 1:** Write an algebraic expression for “The sum of twice a number ‘x’ and 5.”

**Answer:** 2x + 5

**Problem 2:** Translate the following phrase into an algebraic expression: “Five less than three times a number ‘y’.”

**Answer:** 3y – 5

**Problem 3:** Create an algebraic expression for “The product of a number ‘a’ and its square minus 7.”

**Answer:** a^2 – 7

**Problem 4:** Write an algebraic expression for “One-third of a number ‘m’ increased by 4.”

**Answer:** (1/3)m + 4

**Problem 5:** Translate this statement into an algebraic expression: “The difference between twice the sum of ‘p’ and ‘q’ and 10.”

**Answer:** 2(p + q) – 10

**Problem 6:** Translate “ten less than the product of a number and two” into an algebraic expression.

**Answer:** Let ‘p’ represent the number. The expression is 2p – 10.

**Problem 7:** Write an algebraic expression for “The absolute value of the difference between a number ‘x’ and 6.”

**Answer:** |x – 6|

**Problem 8:** Translate this phrase into an algebraic expression: “Three times the quantity of ‘a’ plus half of ‘b’.”

**Answer:** 3a + (1/2)b

**Problem 9:** Translate “the product of three times a number and two less than the number” into an algebraic expression.

**Answer:** Let ‘r’ represent the number. The expression is 3r(2r – 2).

**Problem 10:** Translate the following sentence into an algebraic expression: “Twice the sum of ‘m’ and ‘n’ minus ‘p’.”

**Answer:** 2(m + n) – p

These practice problems cover a range of algebraic expressions, including addition, subtraction, multiplication, division, exponents, absolute value and more. Use these to sharpen your algebraic expression skills.

## Types of algebraic expressions

There are 4 types of algebraic expressions – ** Polynomial,Trinomial ,Binomial ,Monomial**.

### Which algebraic expression is a polynomial

A polynomial is an algebraic expression consisting of variables, coefficients, and non-negative integer exponents, combined using addition and subtraction, but not division by variables. Here are some algebraic expressions, and I’ll identify which ones are polynomials:

**3x^2 + 5x – 2**

This is a polynomial. It consists of terms with variables (x), coefficients (3, 5, -2), and non-negative integer exponents (2, 1, 0). It uses addition and subtraction.

**2x^(1/2) + 7**

This is not a polynomial. The exponent is a rational number (1/2), which is not a non-negative integer. Polynomials only allow non-negative integer exponents.

**4x^3y^2 – 6x^2 + 2y**

This is a polynomial. It contains terms with variables (x, y), coefficients (4, -6, 2), and non-negative integer exponents (3, 2, 1, 0). It uses addition and subtraction.

**(2x + 3)/(x – 1)**

This is not a polynomial. It involves division by a variable (x), which is not allowed in polynomial expressions.

**5x^4 – x^3 + 2x^2 – x + 7**

This is a polynomial. It consists of terms with variables (x), coefficients (5, -1, 2, -1, 7), and non-negative integer exponents (4, 3, 2, 1, 0). It uses addition and subtraction.

### Which algebraic expression is a trinomial

A trinomial is an algebraic expression that consists of exactly three unlike terms separated by addition or subtraction. Here are three examples of trinomials:

2x^2 – 3x + 1

In this expression, there are three unlike terms: 2x^2, -3x, and 1, separated by subtraction.

4a^3 + 5a^2 – 2a

There are three unlike terms: 4a^3, 5a^2, and -2a, separated by addition and subtraction.

x^2 – 7xy + 9y^2

This is another trinomial with three unlike terms: x^2, -7xy, and 9y^2, separated by addition and subtraction.

These expressions each have three terms, meeting the definition of a trinomial.

### Which algebraic expression is a Binomial

A binomial algebraic expression is one that consists of exactly two terms, separated by either addition or subtraction.

**3x + 5**

This is a binomial expression with two terms: 3x and 5, connected by addition.

**2y – 7**

This is another binomial expression with two terms: 2y and 7, connected by subtraction.

**a^2 + b**

Here, a^2 and b are two terms connected by addition, making it a binomial expression.

**4z – 2x**

This is a binomial expression with two terms: 4z and 2x, connected by subtraction.

**5m + 3n^2**

This is a binomial expression with two terms: 5m and 3n^2, connected by addition.

In each of these examples, you can see that there are exactly two terms, and they are connected either by addition (+) or subtraction (-), which defines them as binomial expressions.

### Which algebraic expression is a Monomial

A monomial is an algebraic expression that consists of a single term. A term is a combination of a constant and variables, multiplied together. Here are examples of algebraic expressions, and I’ll indicate which one is a monomial:

**3x^2y**: This is a monomial. It’s a single term with variables ‘x’ and ‘y’ raised to powers, all multiplied by a constant (3).

**5xy – 2**: This is not a monomial. It’s a binomial because it has two terms, ‘5xy’ and ‘-2’.

**4z**: This is a monomial. It’s a single term with the variable ‘z’ multiplied by a constant (4).

**2a^2b – 3ab^2 + 1**: This is not a monomial. It’s a trinomial because it has three terms, ‘2a^2b,’ ‘-3ab^2,’ and ‘1’.

## Algebraic expression worksheet

**Monomials:**

Write a monomial for “five times a number.”

Expression: 5x

Express “twice the square of a variable” as a monomial.

Expression: 2x^2

**Binomials:**

Write a binomial for “the sum of a number and four.”

Expression: x + 4

Express “three less than a variable” as a binomial.

Expression: x – 3

**Trinomials:**

Write a trinomial for “the square of a variable plus twice the variable plus one.”

Expression: x^2 + 2x + 1

Express “four times the square of a variable minus three times the variable plus two” as a trinomial.

Expression: 4x^2 – 3x + 2

**Polynomials:**

Write a polynomial for “the sum of the squares of three different variables.”

Expression: x^2 + y^2 + z^2

Express “two times the cube of a variable plus three times the square of the same variable minus five times the variable” as a polynomial.

Expression: 2x^3 + 3x^2 – 5x

**Problems**

Classify the following expressions as monomials, binomials, trinomials, or polynomials: a) 3x^2 – 2x + 1 b) 5y c) 4x^3 – 2x^2 d) 7z^2 – 9

**Answers:** a) Trinomial b) Monomial c) Binomial d) Binomial

Simplify the following expressions: a) 2x^2 – 3x^2 + 5x^2 b) 3y^3 – 2y^3 c) 4z^2 + 7z^2 – z^2

**Answers:** a) 4x^2 b) y^3 c) 10z^2

Write the following expressions in standard form: a) 2x^3 – x^2 + 4x – 3x^3 b) 3y^2 + 5y^2 – 2y^2 c) 6z^4 – 8z^4 + 10z^3

**Answers:** a) -x^2 + 4x b) 6y^2 c) 10z^3 – 2z^4

Add or subtract the following expressions: a) (2x^2 + 3x) + (5x^2 – 2x) b) (4y^3 – 2y) – (3y^3 + y) c) (7z^2 + 3z^3) – (2z^2 – 5z^3)

**Answers:** a) 7x^2 + x b) y^3 – 3y c) 5z^3 + 5z^2

Multiply the following expressions: a) (2x)(3x^2) b) (4y^2)(-2y^3) c) (5z^3)(-2z)

**Answers:** a) 6x^3 b) -8y^5 c) -10z^4

Evaluate the following expressions for x = 2, y = -3, and z = 5: a) 4x^2 – 3y b) 2z^3 + x c) y^2 – 2z

**Answers:** a) 25 b) 42 c) 31

Factor the following expressions: a) 4x^2 – 9 b) x^2 – 25 c) 9y^2 – 16

**Answers:** a) (2x + 3)(2x – 3) b) (x + 5)(x – 5) c) (3y + 4)(3y – 4)

Find the product of the following binomials: a) (2x + 3)(3x – 4) b) (5y – 2)(y + 7) c) (4z + 1)(2z – 5)

**Answers:** a) 6x^2 – 5x – 12 b) 5y^2 + 33y – 14 c) 8z^2 – 6z – 5