Chapter 2 — Algebra

Introduction and motivation: Complex numbers extend the real numbers by introducing the unit i with i^2 = -1. They allow algebraic equations like x^2 + 1 = 0 to have solutions, and they provide a complete theory in which every non-constant polynomial has a root. In algebra and geometry they simplify many calculations and give elegant geometric interpretations.

Algebraic form and basic operations: Write a complex number as z = x + iy where x and y are real numbers; x is called the real part Re(z) and y the imaginary part Im(z). Addition and subtraction are performed component-wise: (x+iy) ± (u+iv) = (x±u) + i(y±v). Multiplication follows distributive law using i^2 = -1: (x+iy)(u+iv) = (xu - yv) + i(xv + yu). Division uses the conjugate: to divide by z2 multiply numerator and denominator by conjugate of z2, giving a real denominator equal to |z2|^2.

Conjugate and modulus: The conjugate of z is the number x - iy. It is useful because z times its conjugate equals x^2 + y^2, which is |z|^2 where |z| is the modulus. The modulus |z| measures distance from origin in the complex plane and is given by sqrt(x^2 + y^2). Note that |zw| = |z||w| and |z+w| ≤ |z| + |w| (triangle inequality).

Polar form and argument: Every non-zero complex number can be written in polar form z = r(cos θ + i sin θ), where r = |z| > 0 and θ is an argument arg(z) measured from the positive real axis. This representation separates magnitude and direction. Use the relations x = r cos θ, y = r sin θ and tan θ = y/x with quadrant check. The principal argument Arg(z) is often chosen in (-π, π] or [0,2π).

Euler’s formula and De Moivre’s theorem: Using exponential notation z = r e^{iθ} compactly encodes polar form. De Moivre’s theorem states (r e^{iθ})^n = r^n e^{inθ} for integer n. This simplifies raising complex numbers to powers and taking powers leads to simple multiplication of arguments and powers of moduli.

Roots of complex numbers and geometry: To find nth roots of a complex number r e^{iθ} write solutions as r^{1/n} e^{i(θ+2kπ)/n} for k = 0,1,...,n-1. The roots are equally spaced around a circle of radius r^{1/n} in the Argand plane. This geometric picture explains why an nth degree polynomial has n complex roots counting multiplicity.

Applications and cautions: Complex conjugation relates to real coefficients: non-real roots of polynomials with real coefficients occur in conjugate pairs. When manipulating arguments and logarithms of complex numbers, remember multi-valued nature of argument: add 2πk when needed. Use complex methods to simplify trigonometric identities, sums and to solve certain algebraic equations that lack real roots.

General form and definitions: A polynomial P(x) of degree n is P(x)=a_n x^n + a_{n-1} x^{n-1} + ... + a_0 with a_n ≠ 0. A root (zero) r satisfies P(r)=0. Over the complex numbers a polynomial of degree n has exactly n roots counting multiplicity. Knowing how roots relate to coefficients allows many conclusions without explicitly solving the polynomial.

Factor form and symmetric sums: If r_1, r_2, ..., r_n are roots then P(x)=a_n ∏_{i=1}^n (x - r_i). Expanding gives coefficients expressed as elementary symmetric sums in the roots. For example the coefficient of x^{n-1} equals -a_n Σ r_i, the coefficient of x^{n-2} equals a_n Σ_{i

Using relations to compute expressions in roots: Many algebraic expressions in roots reduce to functions of symmetric sums. For instance Σ r_i^2 = (Σ r_i)^2 - 2 Σ_{i

Multiple roots and derivatives: A root α is repeated when (x - α)^m divides P(x) for m>1. A repeated root is a common root of P(x) and its derivative P'(x). Thus computing gcd(P, P') detects multiple roots. In exams, remember to check multiplicities especially when using reciprocal or substitution transformations that might introduce or remove multiplicity.

Rational root test and factor theorem: If coefficients are integers, any rational root p/q (in lowest terms) must satisfy p|a_0 and q|a_n. This narrows candidates for rational roots and often finds factors quickly. Factor theorem states (x-α) is a factor iff P(α)=0; use polynomial (or synthetic) division to reduce degree once a factor is found.

Constructing polynomials from given root relations: Sometimes you are asked to form polynomial whose roots are functions of given roots (e.g., squares, reciprocals, sums). Standard method: write x = f(r) for r a root of known polynomial, invert to express r in terms of x (if possible), substitute into original polynomial and clear denominators to obtain polynomial in x. Use symmetric relations and elimination techniques to simplify the resulting polynomial.

Symmetric polynomials and elementary symmetric sums: A polynomial in variables r_1, r_2, ..., r_n is symmetric if swapping any two variables does not change the polynomial. Elementary symmetric sums are fundamental symmetric polynomials: e_1 = Σ r_i, e_2 = Σ_{i

Power sums and Newton’s identities: Define power sums p_k = Σ_{i=1}^n r_i^k. Newton’s identities connect power sums p_k to elementary symmetric sums e_j (or to coefficients of the polynomial). For a monic polynomial x^n + a_1 x^{n-1} + a_2 x^{n-2} + ... + a_n = 0 with roots r_i, the identities give a recursive relation: p_1 + a_1 = 0, p_2 + a_1 p_1 + 2 a_2 = 0, and in general for k ≤ n, p_k + a_1 p_{k-1} + a_2 p_{k-2} + ... + a_{k-1} p_1 + k a_k = 0. For k > n similar relations hold but without the k a_k term and using up to a_n. These formulas let you compute p_k sequentially given the coefficients.

Deriving power sums step by step: To compute p_1 use p_1 = -a_1. Then compute p_2 from p_2 = -a_1 p_1 - 2 a_2. Continue similarly: p_3 = -a_1 p_2 - a_2 p_1 - 3 a_3, and so on until required order. This stepwise approach avoids finding roots explicitly and is particularly effective when roots are complicated radicals but coefficients are simple integers.

Applications in problem solving: Newton’s identities are used when questions ask for Σ r_i^k, Σ 1/r_i, or when forming polynomials whose coefficients depend on power sums. For instance Σ 1/r_i = (Σ_{i

Working with non-monic polynomials and sign conventions: If polynomial is not monic divide by the leading coefficient to put it in monic form before applying identities. Keep careful track of signs: for monic polynomial x^n + a_1 x^{n-1} + ... the relationship a_1 = -e_1, a_2 = e_2, etc., alternates sign. Always verify indexing so that the correct coefficient multiplies the correct power sum in the identity.

Examples and typical exam usage: In board problems you will often compute p_2 or p_3 for cubics and quartics; Newton’s identities give quick answers. Combine these identities with algebraic simplifications to produce neat final values without solving the polynomial. Practice several examples to gain fluency and avoid algebraic sign errors.

Why transform equations? Transformations of the variable are used to simplify polynomials, remove specific terms, reduce to a known solvable form, or normalise coefficients. Common transformations are translation (x = y + c), scaling (x = ky), reciprocal (x = 1/y), and affine transforms x = ay + b. Learning when and how to apply each reduces algebraic complexity and guides solution strategy.

Translation to remove specific terms: A common goal is to make the coefficient of x^{n-1} zero to obtain a 'depressed' polynomial. For a general polynomial a_n x^n + a_{n-1} x^{n-1} + ... choose x = y - a_{n-1}/(n a_n). Substituting and expanding cancels the y^{n-1} term. This is particularly useful for cubic and quartic equations because removing the second-highest power simplifies later methods (Cardano for cubics, Ferrari for quartics) and reduces the number of variables to manage.

Scaling to simplify coefficients and roots: If coefficients are large or if a particular ratio is convenient, set x = ky so the leading coefficient can be normalised or coefficients simplified. Scaling changes the roots by factor k: if x_i are roots of original polynomial, then y_i = x_i/k are roots of the scaled polynomial after appropriate factor adjustment. Use scaling to reduce arithmetic or to bring polynomial into monic form by dividing by a_n after substitution.

Reciprocal transformation: Replace x by 1/y to study reciprocals of roots. The reciprocal polynomial x^n P(1/x) has roots 1/α where α are roots of P(x) (excluding zero). This transformation is convenient when the polynomial is palindromic or when questions ask for sums of reciprocals of roots. Remember to handle roots at zero separately since reciprocal is undefined.

Composite (affine) transformations: The affine substitution x = a y + b combines scaling and translation. Use when mapping a particular value to zero or simplifying a constant term. For example mapping one root to zero may make factorisation easier. After substitution expand carefully and, if needed, divide by common factors to simplify.

Effect on coefficients and process: Always perform substitution and expand fully, collect like powers and then, if desired, divide through by a common factor to obtain a reduced or monic polynomial. Track how p and q in depressed forms are expressed in terms of original coefficients; errors in algebra here cause wrong downstream computations. For cubics and quartics compute explicit formulas for p and q when practising Cardano or Ferrari to get comfortable with symbolic manipulation.

Practical advice and examples: Before applying heavy methods, try rational root test or factor by inspection; sometimes simple factorisation avoids transformations. Use translations when the coefficient of x^{n-1} is non-zero and you seek depressed form; use reciprocal transformation for palindromic polynomials or when asked about reciprocals. After solving in transformed variable, always transform back to original variable and check solutions against domain restrictions.

Reciprocal polynomial definition and basic properties: Given P(x)=a_0 + a_1 x + ... + a_n x^n, the reciprocal polynomial is x^n P(1/x) = a_n + a_{n-1} x + ... + a_0 x^n. If α is a non-zero root of P(x), then 1/α is a root of the reciprocal polynomial. Studying the reciprocal relates roots and their reciprocals and is useful when problems ask about sums or products of reciprocal roots.

Palindromic (self-reciprocal) polynomials: A polynomial is palindromic if its coefficients read the same forwards and backwards: a_k = a_{n-k} for all k. For such polynomials P(x) satisfies x^n P(1/x) = P(x) up to a constant multiple. This symmetry implies if α is a root then 1/α is also a root (with same multiplicity), so roots come in reciprocal pairs unless α = ±1 or α = 0 where special cases occur.

Antipalindromic polynomials: If a_k = -a_{n-k}, the polynomial satisfies x^n P(1/x) = -P(x). In this case roots again occur in reciprocal pairs with sign relationships. Observe special behaviour when degree is odd or when particular roots like ±1 occur.

Reduction by substitution y = x + 1/x: Many palindromic polynomials, especially those with only even powers or symmetric in x and 1/x, can be reduced by setting y = x + 1/x. Use identities x^2 + 1/x^2 = y^2 - 2, x^3 + 1/x^3 = y^3 - 3y, and so on to express symmetric combinations in terms of y. This reduces degree and leads to a polynomial in y of lower order, which is easier to solve. After finding y-values, solve x + 1/x = y to recover x by solving quadratic x^2 - y x + 1 = 0.

Handling zero roots and multiplicities: If P(0)=0 then zero is a root and reciprocal transformation is not directly applicable. Remove factors of x first, then consider reciprocal polynomial of remaining factor. For palindromic polynomials if degree is even they may factor into quadratics in x + 1/x or into reciprocal factor pairs; if degree is odd check for root x = -1 or x = 1 explicitly.

Practical steps and examples: First check coefficient symmetry to identify palindromic or antipalindromic structure. If symmetric, try substitution y = x + 1/x or form x^n P(1/x) and compare to P(x) to exploit reciprocity. Use reciprocal polynomials to turn problems about reciprocals into direct root questions. In examinations, recognising palindromic structure quickly saves time and avoids heavy algebraic factorisation.

Reducing the general cubic: A general cubic ax^3 + bx^2 + cx + d = 0 can be simplified by eliminating the quadratic term. Use translation x = y - b/(3a). Substituting and expanding produces a depressed cubic of the form y^3 + p y + q = 0, where p and q are expressed in terms of a, b, c, d. This removal of the quadratic term reduces complexity and makes the cubic amenable to solution by radicals.

Formulas for p and q: After substitution compute p = (3ac - b^2)/(3a^2) and q = (2b^3 - 9abc + 27a^2 d)/(27 a^3). These expressions arise from algebraic expansion; practice computing them symbolically to avoid arithmetic mistakes. The reduced cubic y^3 + p y + q is the canonical form for applying Cardano’s method.

Cardano’s method and discriminant: Define Δ = (q/2)^2 + (p/3)^3. If Δ>0 there is one real root and two complex conjugate roots. If Δ=0 there are multiple roots; if Δ<0 there are three distinct real roots. Cardano’s formula produces a solution y = ∛(-q/2 + √Δ) + ∛(-q/2 - √Δ). Care is needed when Δ<0 because the square root inside Cardano is imaginary; in that case use trigonometric form to express the three real roots.

Trigonometric form for three real roots: When Δ<0 set r = √(-p^3/27) and compute cos 3θ = (-q/2)/r. Then y_k = 2 √(-p/3) cos((θ + 2kπ)/3) for k = 0,1,2 gives three real solutions. This form avoids taking cube roots of complex numbers explicitly and is numerically stable for real roots.

Branch choices and complex cube roots: Cardano involves cube roots of complex numbers; there are three cube roots of any complex number and choosing different branches yields the three roots of cubic. For computations when Δ>0, one real cube root combination gives the real root directly. When Δ≤0, prefer trigonometric expressions to list real roots clearly.

Practical strategy and checks: Before applying Cardano always test for simple rational roots using rational root test; if found factor cubic and reduce to quadratic. After finding candidate roots, substitute back into original equation to verify. For exam answers show reduction to depressed cubic, computation of p, q, Δ, and then give explicit formulas or trigonometric forms depending on sign of Δ. State any domain or branch choices used and simplify radicals where possible.

Overview and first reduction: A quartic equation ax^4 + bx^3 + cx^2 + dx + e = 0 may be simplified by first translating x = y - b/(4a) to eliminate the cubic term. The depressed quartic then has the form y^4 + p y^2 + q y + r = 0. This reduction reduces algebraic complexity and sets the stage for Ferrari’s method or other factorisation strategies.

Ferrari’s idea: Ferrari’s method introduces an auxiliary parameter to express the depressed quartic as a difference of squares, allowing factorisation into two quadratics. Specifically, add and subtract a suitable square (y^2 + α)^2 and choose α so that the quartic becomes (y^2 + α)^2 - (β y + γ)^2; then it factors into (y^2 + α + β y + γ)(y^2 + α - β y - γ). Matching coefficients and eliminating the auxiliary variables leads to a resolvent cubic whose solution yields α, β, γ and hence factors of the quartic.

Resolvent cubic and solution path: Equating coefficients after the square manipulation produces relationships between α, β, γ and p, q, r. Eliminating β and γ leaves a cubic equation in α (or in u = α^2 depending on derivation). Solve this resolvent cubic (by factorisation or Cardano) and pick the appropriate real root that leads to real factorisation. Once α is known compute β and γ and solve the resulting quadratic factors to obtain the four roots of the original quartic.

Special quartics and shortcuts: Some quartics are biquadratic (only even powers) and reduce to a quadratic in x^2 using substitution z = x^2. Palindromic quartics can be reduced using substitution x + 1/x. Where rational roots exist use rational root test to factor out linear terms first. In many examination problems the quartic is chosen to allow a clean algebraic factorisation with small integers.

Practical advice and verification: Ferrari’s method involves heavy algebra; perform steps carefully and show reasoning when presenting in answers. After obtaining roots, verify by substitution into original equation, especially if you selected one root of the resolvent cubic among several possible roots. When coefficients are real, complex roots occur in conjugate pairs; record their conjugation when presenting final answers.

Why inequalities matter: Inequalities are essential in algebra for establishing bounds, proving optimal values and constraints, and solving extremal problems. They appear frequently in competitive and board exams, and they provide tools to control complicated expressions by simpler quantities.

AM-GM inequality: For non-negative real numbers a_1, a_2, ..., a_n, the arithmetic mean is at least the geometric mean: (a_1 + ... + a_n)/n ≥ (a_1 a_2 ... a_n)^{1/n}. Equality holds when all a_i are equal. Use AM-GM to derive minimum or maximum of expressions where product or sum is constrained. For example to minimise x + k/x for x>0, split terms to apply AM-GM.

Proof ideas for AM-GM: AM-GM can be proved by induction, by rearrangement, or by using convexity of logarithm. A simple two-variable case follows from (√a - √b)^2 ≥ 0 leading to (a+b)/2 ≥ √(ab). General case follows by induction or by writing numbers as weighted geometric mean using repeated averaging.

Cauchy-Schwarz inequality: For sequences (u_i) and (v_i) of real numbers, (Σ u_i v_i)^2 ≤ (Σ u_i^2)(Σ v_i^2). This is a central inequality used to bound sums of products. It leads to many corollaries, such as RMS ≥ AM and Titu’s lemma: Σ (a_i^2/b_i) ≥ (Σ a_i)^2 / Σ b_i for positive b_i. Use Cauchy-Schwarz to transform complicated expressions into product of simpler sums, facilitating comparison and bounding.

Rearrangement inequality: If two sequences sorted in same order are paired termwise, their sum of products is maximal; if sorted oppositely, it is minimal. This helps determine optimal pairings in optimisation problems. Use it when you need to maximise or minimise Σ a_i b_{σ(i)} over permutations σ.

Other useful inequalities and techniques: Use (x - y)^2 ≥ 0 to derive x^2 + y^2 ≥ 2xy and more general quadratic inequalities. Jensen’s inequality for convex functions generalises AM-GM and gives bounds for means. Holder and Minkowski are higher-level inequalities useful for more advanced contexts. In algebra problems, combining inequalities cleverly often yields the neatest solution.

Strategy when solving problems: Identify positivity conditions required by the inequality, normalise constants so that AM-GM applies directly, and check equality case to find when the bound is attained. When applying Cauchy-Schwarz choose u_i and v_i to reflect the structure of the expression you wish to bound. For contest problems attempt several inequalities: sometimes AM-GM gives a bound but Cauchy-Schwarz yields a sharper one.

Definition and types of sequences: A sequence is an ordered list a_1, a_2, a_3, ... defined by a rule for the nth term. Sequences can be defined explicitly by a_n = f(n) or recursively by relating a_{n+1} to previous terms. Two most important simple sequences are arithmetic progressions (AP) and geometric progressions (GP).

Arithmetic progression (AP): In an AP consecutive terms differ by a constant d called the common difference. The nth term is a_n = a_1 + (n-1) d. The sum of first n terms S_n = n/2 (2a_1 + (n-1) d) = n(a_1 + a_n)/2. Properties such as the average of symmetric terms being constant and arithmetic mean relations are useful in solving problems about averages, medians and evenly spaced quantities.

Geometric progression (GP): In a GP consecutive terms have constant ratio r, called common ratio. The nth term is a_n = a_1 r^{n-1}. Sum of first n terms is S_n = a_1 (1 - r^n)/(1 - r) for r ≠ 1. If |r| < 1 the infinite series converges to S_∞ = a_1/(1 - r). Use GP to model exponential growth or decay, compound interest and repeated multiplicative processes.

Relations and means: In an AP the arithmetic mean of two terms equals the term midway in the index. In a GP the geometric mean of two terms equals the middle term when indices are symmetric: for terms ar^{k-1}, ar^k, ar^{k+1}, we have (ar^k)^2 = (ar^{k-1})(ar^{k+1}). Arithmetic, geometric and harmonic means are often compared using inequalities like AM-GM and used to solve optimisation problems involving sequences.

Recursive sequences and linear recurrences: Some sequences satisfy linear recurrence relations like a_{n+1} = p a_n + q. Solve such relations by finding a particular solution and adding homogeneous solution; for homogeneous linear recurrence with constant coefficients use characteristic equation. For example Fibonacci-type sequences follow a_{n+2} = a_{n+1} + a_n and are solved via characteristic roots giving closed form (Binet’s formula).

Convergence and limits: Understand when a sequence converges. AP with non-zero d diverges; GP converges if |r|<1. For recursive sequences study fixed points and use monotone convergence tests. Many exam problems ask for limits of sequences or for sums of sequences which require recognising AP or GP structure or converting to such forms.

Problem solving tips: Express complicated terms in known AP/GP form, use S_n formulas, pair terms when helpful and check boundary conditions. For mixed-type sequences try factoring or substitution to reveal underlying AP or GP structure.

Definition and partial sums: A series is the sum of terms of a sequence: for a sequence (a_k) the series is Σ_{k=1}^∞ a_k. The nth partial sum S_n = Σ_{k=1}^n a_k helps study whether the series converges: if S_n approaches a finite limit as n→∞ the series converges to that limit. In algebra problems you often need to evaluate finite sums exactly or decide convergence of simple infinite series.

Telescoping sums and decomposition: Telescoping sums are sums where terms cancel when written as differences. To use this, write a_k as f(k) - f(k+1) or use partial fraction decomposition to produce such differences. For example 1/(k(k+1)) = 1/k - 1/(k+1) so Σ_{k=1}^n 1/(k(k+1)) = 1 - 1/(n+1). Recognising telescoping structure is a key skill that turns long sums into a few boundary terms.

Pairing and symmetry methods: Pair terms such as a_k + a_{n+1-k} when the sequence has symmetry. For arithmetic progressions pairing first and last yields n/2 pairs each summing to same value, leading to formula for S_n. For sums involving binomial coefficients use symmetry of C(n,k) = C(n,n-k) to simplify computations and to derive identities.

Using known series and generating functions: Use geometric series formula Σ_{k=0}^∞ x^k = 1/(1-x) for |x|<1 and manipulate it to obtain other sums by differentiation or multiplication. Generating functions and power series convert combinatorial or algebraic sums into coefficients extraction problems. For example differentiating the geometric series gives Σ_{k=1}^∞ k x^{k-1} = 1/(1-x)^2, which leads to formulas for weighted sums of binomial coefficients.

Binomial identities applied to sums: Binomial coefficients simplify many sums: Σ_{k=0}^n C(n,k) = 2^n, Σ k C(n,k) = n 2^{n-1}, and higher weighted sums follow from differentiating (1 + x)^n. Use these when sums involve combinations or polynomial weights multiplied by binomial coefficients.

Power series and radius of convergence: Power series Σ c_k x^k represent functions within their radius of convergence. Use ratio or root test to determine radius R such that the series converges for |x| < R. Inside the radius you may integrate or differentiate the series termwise. For algebra problems expand functions like (1 + x)^α using generalised binomial theorem for |x|<1 and use first few terms for approximation or exact algebraic manipulation when asked.

Techniques for evaluating finite sums: Convert rational terms using partial fractions to produce telescoping differences, use substitution to reveal geometric or arithmetic structure, and apply combinatorial identities where binomial coefficients appear. When asked for closed-form of S_n show step-by-step decomposition and simplify boundary terms. For infinite sums always state convergence condition and compute limit of partial sums.

Finite binomial theorem and combinatorial meaning: For non-negative integer n, the expansion (1 + x)^n = Σ_{k=0}^n C(n,k) x^k holds where C(n,k) = n!/(k!(n-k)!). Each coefficient counts combinations: C(n,k) is the number of ways to choose k items from n. This combinatorial interpretation helps derive identities and understand properties like symmetry and Pascal’s identity.

Deriving useful identities: Pascal’s identity C(n,k) = C(n-1,k-1) + C(n-1,k) follows from considering whether a particular element is chosen or not. Summing coefficients gives Σ_{k=0}^n C(n,k) = 2^n since each subset of n elements either includes or excludes each element. Weighted sums such as Σ k C(n,k) = n 2^{n-1} follow from differentiating (1 + x)^n and evaluating at x = 1; this algebraic method is often quicker than combinatorial reasoning when handling more complex weights.

Generalised binomial theorem for any exponent: For any real or complex α, (1 + x)^α = Σ_{k=0}^∞ C(α,k) x^k where C(α,k) = α(α-1)...(α-k+1)/k!. This infinite series converges for |x|<1 and reduces to finite sum when α is a non-negative integer. Use this generalised form to expand expressions like (1 + x)^{-1} or (1 + x)^{1/2} into power series and to obtain approximations or exact algebraic manipulations within radius of convergence.

Extracting coefficients and general term: In (a + b)^n the general term is C(n,k) a^{n-k} b^k. To find coefficient of a specific power in a product or composition, represent the expression in binomial form or use generating functions. Many exam problems ask for particular coefficient values or sum of selected coefficients; set up the general term and identify the required power to extract coefficient directly.

Applications and limits: Use the finite theorem for exact algebraic expansions; use the generalised binomial series to expand negative or fractional powers when |x|<1. When using the infinite series always state the domain of convergence and justify termwise operations like differentiation or integration. In algebra problems combine binomial expansions with substitution to evaluate sums, coefficients or limits.

Exam-style techniques and tips: Be fluent with factorial calculations for small values, use Pascal’s triangle for quick coefficients, and know standard binomial sums. When α is not an integer, write the first few terms explicitly and state convergence condition. For coefficient problems, explicitly write the general term and solve for index k that produces required power; include simplification steps to present neat final answers.

Purpose and overall method: Partial fraction decomposition rewrites a proper rational function P(x)/Q(x) with deg P < deg Q as a sum of simpler rational pieces. This simplifies integration, summation, and algebraic manipulation. If deg P ≥ deg Q perform polynomial long division first to separate polynomial part and a proper fraction. Then factor the denominator completely over the reals into linear and irreducible quadratic factors to set up the decomposition.

Forms for decomposition: For distinct linear factors (x - a) include A/(x - a) terms. For repeated linear factors (x - a)^m include A_1/(x-a) + A_2/(x-a)^2 + ... + A_m/(x-a)^m. For irreducible quadratic factors ax^2 + bx + c use (Bx + C)/(ax^2 + bx + c). For repeated quadratic factors include higher powers with similar numerators. Over real numbers keep quadratics unreduced; over complex numbers you may reduce further to linear factors.

Finding constants efficiently: Multiply both sides by the denominator Q(x) to clear fractions, obtaining an identity in x. Solve for constants by plugging convenient x-values (especially roots of Q(x)) and by equating coefficients of like powers of x. For simple distinct linear factors the cover-up method quickly gives residues: A = P(a)/Q'(a) when a is a simple root of Q. For quadratics solve the resulting linear system for unknowns B and C by equating coefficients or evaluating at two distinct x-values.

Telescoping sums and applications: Many algebraic sums have general terms that are rational; partial fractions often convert these to telescoping sums. For example 1/(k(k+1)(k+2)) decomposes into a combination of 1/(k(k+1)) and 1/((k+1)(k+2)) which telescope. Partial fractions also simplify integration of rational functions, reducing them to logarithmic and inverse tangent terms when integrating real quadratics.

Practical considerations and verification: After determining constants recombine the partial fractions to verify equality with original rational function. Keep track of domain restrictions where denominators vanish—these are excluded values. For exam work show division step (if necessary), factorisation of denominator, set-up of decomposition, calculation of constants, and final recombination or telescoping sum evaluation. Detailed intermediate steps often earn method marks even if arithmetic errors occur.

Rational equations: Rational equations contain ratios of polynomials. To solve, first note domain restrictions where denominators are zero. Multiply both sides by the least common denominator to clear fractions, then solve the resulting polynomial equation. Always check candidates in the original equation to reject extraneous roots introduced by multiplication.

Exponential equations: For a^{f(x)} = b^{g(x)} convert to same base if possible. If bases equal and positive not equal to 1, equate exponents. If different, take logarithms: f(x) ln a = g(x) ln b. For equations like a^{g(x)} = h(x) where h(x) is algebraic, try substitution t = a^{x} to reduce to algebraic equations. Remember exponential functions are always positive so any solution must give positive values on both sides.

Logarithmic equations: Use definition log_a x defined for x>0 and a>0, a≠1. Convert log equations to exponential form: log_a f(x) = c ⇔ f(x) = a^c. Use log rules to combine logs: log(uv) = log u + log v, log(u/v) = log u - log v and log(u^r) = r log u. Check domain: all arguments of logarithms must be positive; this often excludes extraneous solutions after algebraic manipulation.

Mixed types and substitution: Many problems mix rational, exponential and logarithmic expressions. A standard tactic is to isolate the exponential/logarithmic part and substitute t for it to obtain an algebraic equation. For example solve 2^{x} + 2^{-x} = 5 by letting t = 2^{x} so t + 1/t = 5, multiply through to get quadratic in t. Similarly transform log sums into single log and exponentiate to solve.

Graphical and numerical considerations: Some equations do not admit algebraic closed-form solutions and require numerical or graphical methods. For board exams problems are designed to be algebraically solvable; still check monotonicity and uniqueness by considering monotone behaviour of exponential and logarithmic functions to justify single solution or multiple solutions.

Practical tips: Check domains before and after operations, prefer substitution to simplify mixed expressions, and test all roots in original equation. When taking logarithms of both sides ensure both sides are positive and handle sign cases separately when necessary.

Definition and notation: A matrix is a rectangular array of numbers with m rows and n columns; denote it as A = [a_{ij}] with i row index and j column index. Matrices represent data, linear transformations and systems of linear equations compactly. Square matrices (n×n) have special algebraic properties like inverses and determinants.

Basic operations: Matrix addition and scalar multiplication are performed entrywise: (A + B)_{ij} = A_{ij} + B_{ij}. Matrix multiplication AB is defined when number of columns of A equals number of rows of B, with (AB)_{ij} = Σ_k A_{ik} B_{kj}. Matrix multiplication is associative and distributive but not generally commutative. The transpose A^T flips rows and columns and satisfies (AB)^T = B^T A^T.

Determinant and its properties: Determinant is a scalar function defined for square matrices that measures, among other things, the scaling factor of area or volume under the associated linear map. For 2×2 matrix [[a,b],[c,d]] det = ad - bc. Determinant properties include det(AB) = det A · det B, det(A^T) = det A, and det(I) = 1. A matrix is invertible iff its determinant is non-zero.

Inverse and solving linear systems: If A is invertible (square and det A ≠ 0) its inverse A^{-1} satisfies AA^{-1} = I. For 2×2 matrix inverse formula is explicit: [[a,b],[c,d]]^{-1} = (1/(ad - bc)) [[d,-b],[-c,a]]. For solving Ax = b when A is invertible the unique solution is x = A^{-1} b. For small systems Cramer’s rule gives x_i = det(A_i)/det(A) where A_i is matrix obtained by replacing ith column by b vector; it demonstrates role of determinants but is practical only for small n due to heavy computation.

Rank and consistency: Rank of a matrix is the maximum number of linearly independent rows or columns. For system Ax = b consistency depends on whether rank([A|b]) equals rank(A). If equal and equals number of unknowns there is a unique solution; if rank less than number of unknowns there are infinitely many solutions with parameters. Use row reduction to echelon form to compute rank and solve systems.

Applications and exam tips: Matrices compactly express linear equations, perform transformations, and represent recurrences. Be comfortable performing elementary row operations, computing determinants by expansion or by reducing to triangular form, and using inverse formula for 2×2 matrices. Show steps when solving systems and check solutions by substitution into original equations.

Definitions and motivation: For a square matrix A, an eigenvalue λ is a scalar such that there exists a non-zero vector v with Av = λ v. The vector v is called an eigenvector corresponding to λ. Eigenpairs (λ, v) reveal invariant directions for the linear transformation represented by A and simplify many computations like raising A to a power.

Characteristic polynomial and computation: Eigenvalues are found by solving det(A - λ I) = 0. For a 2×2 matrix A = [[a,b],[c,d]] the characteristic polynomial is λ^2 - (a + d) λ + (ad - bc). Its roots are eigenvalues; their sum equals trace(A) = a + d and their product equals det(A) = ad - bc. These relations permit quick checks and computations without full root solving in some problems.

Finding eigenvectors: For each eigenvalue λ solve (A - λ I) v = 0 to find non-zero solutions v in the nullspace. In a 2×2 case this reduces to a single linear equation (up to proportionality) yielding eigenvector direction; choose a convenient non-zero component to present a concrete vector. Normalize eigenvectors if required for orthogonality or diagonalisation using orthogonal matrices.

Diagonalisation and its benefits: If A has two linearly independent eigenvectors P = [v_1 v_2] then A = P D P^{-1} with D diagonal containing eigenvalues. Diagonalisation simplifies computing powers: A^n = P D^n P^{-1}, and functions of matrices like e^{tA} reduce to applying the scalar function to diagonal entries. For symmetric real matrices eigenvectors are orthogonal and P can be chosen orthogonal, giving P^{-1} = P^T which simplifies computations and numerical stability.

Conditions and repeated eigenvalues: Distinct eigenvalues guarantee linearly independent eigenvectors for 2×2 matrices and hence diagonalisation. If eigenvalue is repeated check geometric multiplicity (number of independent eigenvectors); if less than algebraic multiplicity the matrix is not diagonalizable and one must use Jordan forms (beyond syllabus). For exam problems often matrices have distinct eigenvalues making diagonalisation straightforward.

Applications and exam practice: Use diagonalisation to compute A^n efficiently and to solve linear recurrences represented by matrix multiplication. In examinations compute characteristic polynomial, find eigenvalues, find associated eigenvectors, form P and D, and verify decomposition. Provide final computations such as A^3 or A^n for small n to demonstrate application.

Algebra as modelling language: Algebra transforms real-world descriptions into equations that can be manipulated. Translating a word problem into algebraic symbols, choosing appropriate substitutions, and applying structural methods like factorisation or matrix formulation is a key applied skill. Algebraic models include linear systems, polynomial approximations, and recurrence relations.

Transformations to simplify models: Use reciprocal or logarithmic transformations to linearise non-linear data, or scaling to normalise units. For instance, fitting an exponential growth pattern y = A r^t can be linearised by taking logarithms: ln y = ln A + t ln r so linear regression or algebraic solution methods become applicable. Translational substitutions map complicated polynomials to depressed forms enabling known solution methods.

Systems and matrices in applications: Systems of linear equations model many practical problems: mixtures, supply-demand, electrical networks and flow problems. Representing systems as Ax = b and using matrix methods (row reduction, inverse, Cramer’s rule) produces solutions efficiently. Matrices also model transformations in geometry, population models with Leslie matrices, and data transformations in applied maths.

Sequences and series in modelling: AP and GP model arithmetic growth (e.g., equal increments) and geometric growth (compound interest, population growth), respectively. Use GP to compute compound interest and AP for linear savings plans. Infinite geometric series model repeated discounting or present value calculations when |r|<1. Recurrence relations model processes where current state depends on past states and can be solved by characteristic equations or matrix powers.

Optimisation and inequalities: Inequalities provide bounds and optimise quantities under constraints without calculus. AM-GM and Cauchy-Schwarz are applied to minimise cost, maximise area or determine extreme values in discrete or continuous settings. Algebraic manipulation combined with inequality knowledge yields solutions amenable to board exam formats.

Approach to problem solving: Read the problem carefully, translate to algebraic form, choose substitutions that reduce complexity, apply known algebraic techniques, and interpret results in the original context. Check units, domain restrictions and reasonableness of answers. Present work in clear steps to ensure method marks even if arithmetic is involved.

Purpose of elimination: When faced with two or more polynomial equations in several variables we often need to eliminate one variable to reduce the system to a single polynomial equation in one variable. This step reduces a multivariable algebraic problem to a univariate problem that can be solved by standard algebraic methods. Resultants provide an algebraic tool to carry out elimination systematically.

Resultant and Sylvester matrix: The resultant Res_x(P,Q) of two polynomials P(x) and Q(x) (viewed as polynomials in x with coefficients that may depend on other parameters like y) is constructed as the determinant of the Sylvester matrix built from coefficients of P and Q. The key property is Res_x(P,Q) = 0 precisely when P and Q have a common root in x. Thus computing the resultant with respect to x eliminates x and yields a polynomial in the remaining parameter(s) whose roots are values for which the system has a solution.

Practical elimination strategy: For a system P(x,y)=0 and Q(x,y)=0 treat them as polynomials in x with coefficients as polynomials in y. Compute the resultant Res_x(P,Q) to obtain a polynomial in y; solve this univariate polynomial to find possible y-values. Substitute each y back into one of the original equations and solve for x. This method is powerful but can be algebraically heavy by hand; problems given in examinations are usually arranged so that elimination simplifies or factors neatly.

Alternative elimination methods: Often direct substitution, factoring, comparing coefficients or using the Euclidean algorithm for polynomials suffices for elimination without forming the full Sylvester determinant. For two equations linear in one variable solve directly; for nonlinear systems examine symmetry or use resultant simplifications by factoring common parts. Use the discriminant to detect multiple roots: discriminant of P is proportional to resultant of P and its derivative P', vanishing when multiple roots exist.

Applications and interpretation: Elimination is used to derive conditions under which two curves intersect, to find polynomial relations between algebraic expressions of roots, and to eliminate parameters in families of polynomials or curves. Resultants link to discriminants and provide a criterion for common factors. In applied problems elimination clarifies compatibility conditions among constraints.

Exam guidance: Understand the concept and the key property of the resultant; practice elimination by substitution and factorisation for typical exam problems. Reserve full Sylvester determinant calculations for small degree polynomials or when structure simplifies the determinant; always check obtained roots by substitution into original equations to discard extraneous solutions introduced by algebraic manipulations.