Chapter 1
Number Systems & Exponents
Laws of Exponents for Real Numbers
a^m × a^n = a^(m+n) | (a^m)/(a^n) = a^(m-n) | (a^m)^n = a^(mn) | (ab)^n = a^n × b^n | a^(-n) = 1/a^n
Fundamental exponential algebra rules for simplifying radicals and roots.
Variables: a > 0 is real, m and n are rational numbers
Exam Tip: a^(p/q) = qth root of (a^p). For example 8^(2/3) = (∛8)² = 2² = 4.
Rationalization of Denominator
1 / (√a + √b) = (√a - √b) / (a - b)
Eliminates irrational roots from the denominator using (x+y)(x-y) = x² - y².
Variables: Multiply numerator and denominator by conjugate pair (√a - √b)
Exam Tip: Always check if denominator simplifies to 1 or -1 before expanding numerators.
Chapter 2
Polynomials & Algebraic Identities
Cubic Algebraic Identities
(a + b)³ = a³ + b³ + 3ab(a + b) | (a - b)³ = a³ - b³ - 3ab(a - b) | a³ + b³ + c³ - 3abc = (a + b + c)(a² + b² + c² - ab - bc - ca)
Vital Olympiad and JEE Foundation identity frequently tested in factorizations.
Variables: Special case: If a + b + c = 0, then a³ + b³ + c³ = 3abc
Exam Tip: Whenever you see x³ + y³ + z³, immediately check if x + y + z = 0! If yes, answer is simply 3xyz.
Remainder & Factor Theorems
Remainder Theorem: p(x) divided by (x - a) gives Remainder = p(a) | Factor Theorem: (x - a) is factor if p(a) = 0
Provides shortcut to test polynomial roots without performing long polynomial division.
Variables: p(x) is polynomial of degree n ≥ 1, a is any real number
Exam Tip: To factorize a cubic polynomial, test x = ±1, ±2 factors of the constant term first.