Algebra

Learning Objectives

• Understand the fundamental concepts of Algebra
• Apply key formulas and techniques to solve problems
• Practice with exam-level questions to build speed and accuracy

Key Concepts

Algebra for Competitive Exams

Algebra deals with mathematical symbols and the rules for manipulating them. Topics include algebraic expressions, identities, linear and quadratic equations, polynomials, sequences and series. Algebra is tested across JEE Main (detailed, high-level), SSC/CGL (intermediate), and CTET (basic upper-primary level).

Key Concepts

Basic Identities: (a+b)² = a²+2ab+b², (a-b)² = a²-2ab+b², (a+b)(a-b) = a²-b², (a+b)³ = a³+3a²b+3ab²+b³, (a-b)³ = a³-3a²b+3ab²-b³, a³+b³ = (a+b)(a²-ab+b²), a³-b³ = (a-b)(a²+ab+b²). Polynomials: Degree, leading coefficient, zeroes. Remainder theorem: if p(x) is divided by (x-a), remainder = p(a). Factor theorem: if p(a)=0, then (x-a) is a factor. Linear Equations: Solving ax+b=0; systems of linear equations in 2 variables (elimination/substitution). Quadratic Equations: ax²+bx+c=0. Roots: x = [-b ± √(b²-4ac)]/2a. Discriminant D=b²-4ac: D>0 real distinct, D=0 real equal, D<0 complex. Sum of roots = -b/a, Product = c/a. Sequences & Series: AP (a, a+d, a+2d...), nth term = a+(n-1)d, sum = n/2[2a+(n-1)d]. GP (a, ar, ar²...), nth term = ar^(n-1), sum = a(r^n-1)/(r-1) for |r|≠1. Permutations & Combinations: nPr = n!/(n-r)!, nCr = n!/[r!(n-r)!], nC0 + nC1 + ... + nCn = 2^n. Binomial Theorem: (1+x)^n = Σ nCr x^r. General term T_(r+1) = nCr x^r.

Solved Examples