Algebraic Expressions And Identities: CBSE Class 8 Maths Chapter 9 Notes

Key Terms in Algebraic Expressions

Algebraic Expression

A combination of constants and variables connected by mathematical operators (+, -, ×, ÷).

Term

The parts of an expression that are separated by + or - signs. A term is a product of its factors.

Coefficient

The numerical factor in a term. For example, in the term 7xy, the coefficient is 7.

Variable

A symbol, usually a letter (like x, y, a), that can represent different numerical values.

Constant

A term in an expression that has a fixed value and does not contain any variables.

Like Terms

Terms that have the same algebraic factors (same variables raised to the same power). Example: 3x²y and -5x²y.

Unlike Terms

Terms that have different algebraic factors. Example: 3xy and 4x²y.

Polynomial

An expression with one or more terms, where the exponents of the variables are non-negative integers.

Identity

An equality that holds true for all possible values of its variables. It is a special type of equation.

Types of Polynomials

How to Add and Subtract Algebraic Expressions

1. — Scan the expressions and group together terms that have the same variables raised to the same powers. (e.g., 5xy and -2xy are like terms; 5x²y and 5xy² are not).
2. — For the column method, write the expressions one below the other, ensuring like terms are in the same column. For subtraction, change the sign of every term in the expression being subtracted.
3. — Add or subtract the numerical coefficients of the like terms. The variable part remains the same. Example: 7x + 3x = (7+3)x = 10x.
4. — Write the resulting terms together to form the final simplified expression. Example: (5x + 3y) + (2x - y) = (5x+2x) + (3y-y) = 7x + 2y.

Understanding Standard Identities

An algebraic identity is a special kind of equation. While a regular equation is true only for certain specific values of its variables, an identity is an equality that remains true for any value you substitute for its variables. Think of them as universal truths in algebra. For example, the equation x + 5 = 8 is only true when x = 3. However, the identity (a + b)² = a² + 2ab + b² is true whether a=1, b=2 or a=10, b=-5. These identities are extremely useful as they provide shortcuts for multiplication and factorization. Instead of performing long multiplication for an expression like (2x + 3y)², you can directly apply the first identity to get the answer quickly. Mastering the four standard identities is essential for solving problems efficiently and is a key skill for more advanced algebra in later classes.

The Four Standard Identities

Multiplication of Expressions: Worked Examples

• {"example":"Multiplying a Monomial by a Binomial","explanation":"Problem: Multiply 3x by (5y + 2)\nSolution: Use the distributive property. Multiply 3x with each term inside the bracket.\n3x (5y + 2) = (3x 5y) + (3x * 2) = 15xy + 6x"}
• {"example":"Multiplying a Binomial by a Binomial","explanation":"Problem: Multiply (2a + 3b) by (a - b)\nSolution: Multiply each term of the first binomial by each term of the second binomial.\n(2a (a-b)) + (3b (a-b))\n= (2aa) - (2ab) + (3ba) - (3bb)\n= 2a² - 2ab + 3ab - 3b²\nCombine like terms (-2ab and +3ab):\n= 2a² + ab - 3b²"}
• {"example":"Using an Identity to Calculate","explanation":"Problem: Calculate 103² using an identity.\nSolution: Write 103 as (100 + 3). This fits Identity I: (a + b)².\nHere, a = 100 and b = 3.\n(100 + 3)² = 100² + 2(100)(3) + 3²\n= 10000 + 600 + 9\n= 10609"}

Must Remember

• {"point":"An expression has no '=' sign, while an equation does."}
• {"point":"Only like terms can be added or subtracted."}
• {"point":"When subtracting expressions, remember to change the sign of EVERY term in the second expression. Example: a - (b - c) = a - b + c."}
• {"point":"The product of two monomials is another monomial."}
• {"point":"The product of a monomial and a binomial is a binomial."}
• {"point":"The product of two binomials can be a binomial, trinomial or have 4 terms, which can be simplified."}
• {"point":"Identities are equations that are true for all values of their variables."}
• {"point":"The degree of a polynomial is the highest power of the variable in the polynomial."}
• {"point":"The coefficient of a term can be positive or negative. Always include the sign."}

Exam Tips and Common Errors

Sign Errors are Costly: The most common mistake is messing up signs, especially during subtraction. When you see -(a - b + c), remember it becomes -a + b - c. Always use brackets to avoid confusion.

Identity Application: Don't just multiply out everything. Examiners want to see if you can apply identities. If you see a number like 99², recognize it as (100-1)² and use Identity II. This saves time and fetches full marks for method.

Like Terms: Double-check your variables and their powers before combining terms. x²y and xy² look similar but are unlike terms and cannot be combined.

Quick Revision Check

• Q: Identify the terms and their coefficients in the expression 5xyz² - 3zy. A: The terms are 5xyz² and -3zy. The coefficient of the first term is 5, and the coefficient of the second term is -3.
• Q: Add the expressions: (7x + 5y) and (2x - 3y). A: (7x + 2x) + (5y - 3y) = 9x + 2y.
• Q: Multiply (p + 8) by (p - 3). A: Using distributive property or Identity IV: p(p-3) + 8(p-3) = p² - 3p + 8p - 24 = p² + 5p - 24.
• Q: Use an identity to find the value of 98 x 102. A: This is in the form (100 - 2)(100 + 2). Using identity (a-b)(a+b) = a² - b², we get 100² - 2² = 10000 - 4 = 9996.

Frequently Asked Questions

Frequently Asked Questions