CBSE Notes Class 7 Chapter 10 Algebraic Expressions

Chapter 10 of CBSE Class 7 Maths, Algebraic Expressions, introduces students to the basics of algebra, focusing on terms, coefficients, and various types of expressions such as monomials, binomials, and polynomials. The chapter explains how to identify like and unlike terms, perform addition and subtraction of expressions, and understand factors and coefficients. It lays the foundation for further algebraic concepts by teaching basic operations on expressions and rules for multiplying and simplifying them, preparing students for higher-level mathematics.

1.0Terms related to Algebraic Expression

A constant is a value that does not change, such as 8, –7, or 0. A variable is a symbol that can take different numerical values, commonly represented by letters like x, y, or z. For example, in the formula for the perimeter of a square, P = 4 x S, 4 is a constant, while P and S are variables.

2.0Algebraic Expression

An algebraic expression is a combination of constants and variables connected by operations (+, –, ×, ÷). For example, in the expression $5-3x+4x^{2}y$, the terms are 5, –3x, and 4x²y. Each term consists of factors: numerical (e.g., 3 in 3x²) and literal (e.g., x, y). For instance, $3x^2$ can be expressed as $3\times x\times x$, and –4xy as $-4\times x\times y$. Unknowns in algebra are represented by letters and form equations.

3.0Factors

Each term in an algebraic expression consists of a product of constants and variables. A numerical factor is a constant, while a literal factor is a variable. For example, in the expression $3x^2-4xy+7y^2$, the terms $3x^2,-4xy$, and $7y^2$ can be broken down into factors: $3x^2=3\times x\times x,-4xy=-4\times x\times y$ and $7y^2=7\times y\times y$. Factors cannot be further simplified, and 1 is not considered a separate factor.

4.0Coefficients 

A coefficient in a term is a numerical or algebraic factor, or their product. For example, in 10xy, 10 is the coefficient of xy, 10x is the coefficient of y, and 10y is the coefficient of x. The numerical coefficient is the number part, while the literal coefficient is the variable part. If no numerical coefficient is specified, it is understood to be 1.

5.0Like and Unlike Terms

Like terms have the same literal factors but can have different numerical coefficients (e.g.,$9x^2,5x^2,and-2x^2;-6x^{2}yand4y{x}^2$ ). 

Unlike terms have different literal factors (e.g., $7x^{2}and9y^2$).

6.0Various Types of Algebraic Expressions

Types of algebraic expressions:

Monomial: Contains one term (e.g., 5x, 2xy, -3a²b).

Binomial: Contains two unlike terms (e.g., 2a + 3b, 8 - 3x).

Trinomial: Contains three terms (e.g., a + 2b + 5c, x + 2y - 3z).

Quadrinomial: Contains four terms (e.g., x + y + z - 5, $x^3+y^3+z^3+3xyz$ ).

Polynomial: Contains one or more terms and follows the form . The degree is the highest power of the variable (e.g., has a degree of 4). For polynomials with multiple variables, the degree is the highest sum of the powers in any term. 

Note: Every polynomial is an expression, but not every expression is a polynomial.

7.0Operations on Algebraic Expressions

1. Addition of Algebraic Expressions

To add algebraic expressions, collect the like terms and add their coefficients. The result is a like term with a coefficient equal to the sum of the original coefficients.

2. Subtraction of Algebraic Expressions

The difference of two like terms is a like term with a coefficient equal to the difference of their numerical coefficients. 

Rule for subtraction: Change the sign of each term in the expression to be subtracted, then add. 

Note: When adding or subtracting expressions, group like terms together or arrange them in columns to make the process easier.

Note: While adding or subtracting algebraic expressions, like terms will be added or subtracted to like terms only.

3. Multiplication of Algebraic Expressions

Before multiplying algebraic expressions, consider these rules:

• The product of two factors with same signs is positive, and with unlike signs is negative.
• For any variable a and positive integers m and n, $a^m\times a^n=a^{m+n}$.
• For example, $x^3\times x^5=x^{3+5}=x^8$.

Rules for Multiplying Monomials:

• The coefficient of the product of two monomials is the product of their individual coefficients.
• The variable part of the product of two monomials is the product of their variables.

4. Division of Algebraic Expressions

Division of Monomials

To divide one algebraic term by another:

• Subtract the exponent of each factor in the denominator from the corresponding factor's exponent in the numerator.
• Divide the numerical coefficients by their HCF.
• If one or both terms are fractions, invert the second term and change the division to multiplication. Simplify by following the above rules or reduce to the lowest terms.

8.0Formulas and Rules Using Algebraic Expressions 

Perimeter formulas

1. Perimeter of an equilateral triangle = $3l$ , where $l$ is the side length.
2. Perimeter of a square = $4l$, where $l$ is the side length.
3. Perimeter of a regular pentagon = $5l$, where $l$ is the side length.

Area formulas:

1. Area of a square = $l^2$, where $l$ is the side length.
2. Area of a rectangle = $l\times b$, where $l$ is the length and b is the breadth.

Area of a triangle = $\frac{1}{2}bh$, where b is the base and h is the height.