Algebra and Functions spans Units P1 to P4 — from basic manipulation through quadratics, polynomials, functions, and the binomial theorem to partial fractions for integration.
Quadratic ax²+bx+c: complete the square to a(x+p)²+q, vertex at (-p, q). Discriminant b²−4ac: >0 two real roots, =0 repeated root, <0 no real roots. Factor theorem: if f(a)=0, then (x−a) is a factor. Remainder theorem: f(a) gives remainder when divided by (x−a). Algebraic division. Partial fractions: A/(x−a) + B/(x−b) for distinct linear factors; A/(x−a) + B/(x−a)² for repeated. Used in integration and series. Binomial expansion: (1+x)ⁿ = 1 + nx + n(n-1)x²/2! + … Valid for |x|<1 when n is not a positive integer.
Function: one-to-one or many-to-one mapping. Domain = inputs; range = outputs. Composite fg(x) = f(g(x)). Inverse f⁻¹: reflect graph in y = x; only exists if f is one-to-one. Modulus |f(x)|: reflects negative parts above x-axis. |f(x)| = a gives f(x) = ±a. Transformations: y = f(x+a) ← left a; y = f(x)+a ↑ a; y = af(x) stretch ×a vertical; y = f(ax) stretch ×(1/a) horizontal; y = −f(x) reflect x-axis; y = f(−x) reflect y-axis. Combinations applied in order.
Partial fractions decompose a rational expression (fraction with polynomials) into simpler fractions. They\'re essential for: integrating rational functions (∫1/((x+1)(x+2))dx is hard, but ∫1/(x+1) − 1/(x+2) dx is easy), and sometimes for series expansions. The binomial expansion (1+x)ⁿ gives a power series approximation when n is fractional or negative, valid for |x|<1. They connect: you might first use partial fractions to split a complex fraction, then apply the binomial expansion to each part separately. For example, to expand (2x+1)/((1+x)(1−x)), first split into partial fractions, then expand each with the binomial theorem.
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