Important Formulas Algebra - & Pedagogy Paper 2 for CTET & TET Exams -

Basic terminology and symbols

• Literal numbers / literals: Letters that represent numbers (for example, x, y, a, b) are called literal numbers or literals.
• Constant: A symbol having a fixed numerical value is called a constant (for example, 2, 5, -3).
• Variable: A letter used to represent a number whose value can vary is called a variable (for example, x, y).
• Term: A single number, a single literal, or a product of numbers and literals (for example, 7, x, 5x, 3ab) is called a term.
• Coefficient: The numerical factor of a term containing a literal is called its coefficient (in 5x, 5 is the coefficient of x).
• Expression: A combination of terms by addition or subtraction (for example, 2x + 3, a + b - 7) is called an algebraic expression.
• Monomial / Binomial / Polynomial: An expression with one term is a monomial; two terms form a binomial; three or more terms form a polynomial.

Rules for writing and interpreting literal expressions

• When letters and numbers are written together without a sign, it means multiplication. Example: 5x means 5 × x.
• Order of factors does not change the product. Example: x × y = y × x = xy.
• Multiplication by 1 leaves a literal unchanged: 1 × x = x.
• Multiplication of a literal by a number may be written with the number before or after the literal: x × 3 = 3x.
• Repeated multiplication of the same literal is written using powers. For example, multiplying a by itself 12 times is written a¹²; multiplying y by itself 15 times is y¹⁵.
• In a power such as x⁹, the letter x is the base and 9 is the index or exponent.

Basic laws of exponents (indices)

• Product of powers with the same base: a^m × aⁿ = a^m+n, provided a = 0.
• Power of a power: (a^m)ⁿ = a^m×n.
• Power of a product: (ab)ⁿ = aⁿ bⁿ.
• Division of powers with the same base: a^m ÷ aⁿ = a^m-n, for m ≥ n and a = 0.
• Zero exponent: a⁰ = 1, for a = 0.
• Negative exponents (extension): a^-n = 1 / aⁿ, for a = 0. (Introduce carefully when students are ready.)

Combining like terms and simplification

• Like terms: Terms that have the same literal part (same variables raised to the same powers) are called like terms. Only like terms can be added or subtracted by combining their coefficients. Example: 5x and -2x are like terms; 3x and 3x² are not like terms.
• Adding like terms: 5x + 3x = (5 + 3)x = 8x.
• Subtracting like terms: 7a - 2a = (7 - 2)a = 5a.
• Multiplying terms: Multiply coefficients and add exponents of like bases. Example: 5x × 3x = (5 × 3) x¹⁺¹ = 15x².
• Dividing terms with same base: Divide coefficients and subtract exponents. Example: 12x⁵ ÷ 3x² = (12 ÷ 3) x^5-2 = 4x³.

Examples with step‐by‐step solutions

Example 1: Simplify 5x + 3x - 2x

Sol. Combine the coefficients of like terms because all terms are multiples of x.

5x + 3x - 2x = (5 + 3 - 2)x

= 6x

Example 2: Simplify 5x × 3x

Sol. Write the product of coefficients, then add the exponents of x.

5x × 3x = (5 × 3) x¹⁺¹

= 15x²

Example 3: Simplify (a × a × a × a) and express using exponent

Sol. Count how many times a is multiplied by itself and write that number as an exponent.

a × a × a × a = a⁴

Example 4: Simplify 12x⁵ ÷ 3x²

Sol.Divide the numerical coefficients and subtract the exponents of x.

12x⁵ ÷ 3x² = (12 ÷ 3) x^5-2

= 4x³

Example 5: Use power rules: (2a)³

Sol. Apply the power of a product rule: raise each factor to the power.

(2a)³ = 2³ a³

= 8a³

Common mistakes to address when teaching

• Do not combine unlike terms (for example, do not add x and x²).
• Remember that the exponent applies only to the literal immediately before it, unless parentheses show otherwise: x²y means x² × y, while (xy)² means x²y².
• Zero exponent rule applies only to nonzero bases: a⁰ = 1 when a = 0.
• When simplifying expressions with both numbers and literals, always keep track of coefficients separately from variable parts.

Teaching tips for building understanding

• Start with concrete examples using repeated multiplication so students see why exponents are a short form for repeated multiplication.
• Use number examples first (for example, 2 × 2 × 2 = 2³ = 8), then replace numbers with letters.
• Emphasise the difference between coefficients and variables by practising combining like terms frequently.
• Encourage students to write each step clearly when simplifying expressions so that the algebraic rules are visible in their work.