Romania Clasa a XII-a Mathematics M1

Chapters: 3

1. Elements of algebra

Groups · Rings and fields · Polynomial rings

Group Theory: Binary Operations, Groups, Rings and Fields – A group is a set with one operation that is closed, associative, has an identity and gives every element an inverse. Clock arithmetic Z6 shows all of it: tables, subgroups, the order of an element and Lagrange's theorem. Rings and fields add a second operation.
Polynomial Division, Horner's Scheme and Roots – A polynomial is a sum of terms like a·xⁿ. You can add and multiply polynomials, and divide them with a remainder: p = q·d + r, where r has a smaller degree than d. Dividing by (x − a) is quick with Horner's scheme, and the remainder is exactly p(a) (Bézout's theorem). If p(a) = 0, then a is a root and (x − a) is a factor. Roots and coefficients are tied together by the Viète relations.

Antiderivatives · Definite integral · Applications of integrals

Integrals – Integration undoes differentiation: if F′(x) = f(x), then ∫f(x) dx = F(x) + C. We integrate with standard formulas, substitution (replace an inside part by u), partial fractions (split a fraction) and by parts (∫u dv = uv − ∫v du). A definite integral ∫ₐᵇ f(x) dx is the signed area under the curve from a to b, and the fundamental theorem says it equals F(b) − F(a). Its properties make many hard integrals easy.
Application of Integrals: Area Under Curves – The area between a curve y = f(x), the x-axis and the lines x = a and x = b is ∫ₐᵇ |f(x)| dx: we add thin vertical strips of height y and width dx. For curves given as x = g(y) we use horizontal strips. Symmetry saves work: find one part of a circle, parabola or ellipse and multiply. A circle of radius r gives πr² and an ellipse with semi-axes a, b gives πab.

Revision for the Bacalaureat

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