What you’ll learn7 learning objectivesChoose one objective for a focused lesson, or study the complete topic.—AHL 1.10 (HL)—Counting and extended binomial theorem• Use counting principles, permutations and combinations.• Extend binomial expansion to fractional and negative indices.• Exclude identical-object permutations, circular arrangements and proof of the theorem.Syllabus objective—AHL 1.11 (HL)—Partial fractions• Decompose rational expressions into partial fractions.• Denominators have at most two distinct linear terms; numerator degree is lower than denominator degree.Syllabus objective—AHL 1.12 (HL)—Complex numbers• Use i where i^2 = -1 and Cartesian form z=a+bi.• Know real/imaginary parts, conjugate, modulus, argument and the Argand diagram.Syllabus objective—AHL 1.13 (HL)—Polar and Euler forms• Use polar form z=r(cos theta + i sin theta)=r cis theta and Euler form z=re^(i theta).• Convert between Cartesian, polar and Euler forms; interpret sums, products and quotients geometrically.Syllabus objective—AHL 1.14 (HL)—Complex roots and De Moivre's theorem• Know complex conjugate roots of real-coefficient polynomial equations.• Use De Moivre's theorem for powers and roots of complex numbers.• Include induction proof for positive integer powers; be aware of rational exponent extension.Syllabus objective—AHL 1.15 (HL)—Advanced proof• Use proof by mathematical induction and proof by contradiction.• Use counterexamples to disprove statements, with explanation of why the counterexample works.Syllabus objective—AHL 1.16 (HL)—Systems of linear equations• Solve up to three linear equations in three unknowns algebraically and with technology.• Identify unique, infinite or no solutions; find general solutions where appropriate.Syllabus objective