Algebra - Term

• Introduction
• What do you mean by an Algebraic Expression?
• Defining Terms in Mathematics
• Components of a Term
• Types of Terms in Algebra
• Combining Terms
• Importance of Terms in Algebra
• Solved Examples
• Practice Problems
• Frequently Asked Questions

Introduction

We use mathematical expressions extensively in Algebra. Expressions are mainly made of terms and operations connecting those terms.

What do you mean by an Algebraic Expression?

An algebraic expression is a math sentence that consists of one or more terms. These terms are connected by mathematical operations such as addition, subtraction, multiplication, or division. 

Defining Terms in Mathematics

A term in a mathematical expression may include numbers, variables, or a combination of both.

Components of a Term

To understand terms better, let's understand their components individually:

• Coefficient: The multiplier of the variable(s) in a term can be represented by a numerical coefficient. For example, in the expression `"3x + 5y - 9"`, the coefficient of the first and second terms are `3` and `5`respectively. A coefficient is placed before the variable in a term.
• Variable: A variable is a symbol (typically a letter) that denotes an unknown quantity. In `3x + 5y - 9`, the variables used in the expression are `x` and `y`.
• Constants: Constants are numbers that do not have variables associated with them and are also considered as terms. For instance, in the expression `3x + 5y - 9`, the constant is `-9`.

• Exponent: Exponents represent the power to which a variable is raised. For example, the exponent in `4x²` is `2`.

• Factors: The numbers that are multiplied to create a term are called factors. For example, in the term `"2x - 3"`, the factors of the first term `2x` are `2` and `x`.

Types of Terms in Algebra

In algebra, terms can be divided into two categories: 

• Like Terms 
• Unlike Terms

Like Terms: Terms with the same variables raised to the same exponent are said to be "like terms." We can add or subtract like terms. 

For example: `2x and 9x` are like terms because both the terms have the common variable `x`. Also, variable `x` in both terms is raised to the exponent `1`.

Unlike Terms: Terms with different variables and / or exponents are said to be "unlike terms."  We cannot add or subtract unlike terms.

For example:  `4w` and `10w²` are not like terms though they share the same variable. This is because in the first term `4w, w` is raised to exponent `1` and in the second term `10w²`, w is raised to exponent `2`.

Combining Terms

Algebraic expressions are built by combining various terms using mathematical operations. 

For example, consider the expression `3x + 2y - 5` In this expression, `3x,` `2y,` and `5` are individual terms combined with addition and subtraction.

Importance of Terms in Algebra

Terms are crucial in algebra for several reasons:

• Simplification: Terms can be manipulated and simplified using algebraic operations. For example, you can factor out common factors, combine like terms, or expand expressions to help solve equations.
• Equation Solving: Equations often involve terms on both sides, and understanding how terms interact is essential for solving equations.
• Polynomial Functions: Polynomial functions are built by combining terms. 
• Classifying polynomials: Algebraic expressions or polynomials can be classified as monomials, binomials, trinomials or polynomials depending on the number of terms in the expression.

Solved Examples

Example 1: Determine the constant, variable, and terms in the `5w + y + 9` ?

Answer: In the given algebraic expression, we have:

Example 2: Determine the factors in the algebraic expression `15wxyz` .

Answer: In the given algebraic expression, the factors are `15, w, x, y,` and `z`.

Example 3: Match the following-

Answer: The correct match is-

Practice Problems

Q1. Which of the following expressions have like terms?

1. `15w + 17y`
2. `10y + 5y`
3. `12w + 9z`
4. `14x + 8y`

Answer: b

Q2. Find terms in the algebraic expression `7wy + 8` ?

1. `7, w, 8`
2. `w, y, 8`
3. `7wy, 8`
4. None of these

Answer: c

Q`3`. Consider the expression \(2x^3 - 5x^2y + 7z\). Identify the type of expression from the options.

1. Monomial
2. Binomial
3. Trinomial
4. Quadrinomial

Answer: c

Q4. Determine the number of terms in the expression \(4a^2 - 3ab + 7b - 2\).

1. `2`
2. `3`
3. `4`
4. `5`

Answer: c

Frequently Asked Questions

Q`1`. What are the different types of polynomials based on the number of terms they have?

Answer: Monomials, binomials, and trinomials are different types of polynomials with one, two, and three terms respectively. Expressions with more than `3` terms are generically called polynomials.

Q`2`. What is the basic difference between like terms and unlike terms?

Answer: Like terms have the same variable with variables raised to the same exponent while unlike terms may have different variables or the same variables raised to different exponent values.

Q`3`. What is an algebraic term?

Answer: An algebraic term is a mathematical expression consisting of constants, variables, and their respective powers and coefficients, combined using arithmetic operations. 

Q`4`. Differentiate between a monomial and a binomial.

Answer: A monomial is an algebraic expression with only one term, which can include constants, variables, or both, combined by multiplication. Examples include \(3x\), \(2y^2\), and \(-5\).

On the other hand, a binomial is an algebraic expression with exactly two terms separated by either addition or subtraction. Examples include \(4x + 2\), \(a^2 - b\), and \(3xy + 7\).