Part I — Symbolic Manipulation

Core Factoring Techniques

Completing the Square

Completing the square converts ax^2 + bx + c into vertex form a(x - h)^2 + k, revealing the parabola's vertex and providing a path to the quadratic formula.

Worked example. Rewrite 2x^2 + 12x + 7 in vertex form.

Step 1: Factor the leading coefficient from the first two terms. 2(x^2 + 6x) + 7

Step 2: Inside the parentheses, add and subtract (6/2)^2 = 9. 2(x^2 + 6x + 9 - 9) + 7

Step 3: Separate the perfect square and distribute. 2(x + 3)^2 - 18 + 7 = 2(x + 3)^2 - 11

The vertex is at (-3, -11) and the parabola opens upward.

Deriving the Quadratic Formula

Completing the square on the general quadratic ax^2 + bx + c = 0 (with a != 0) yields the quadratic formula:

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

Derivation. Start with ax^2 + bx + c = 0. Divide by a: x^2 + (b/a)x + c/a = 0. Move the constant: x^2 + (b/a)x = -c/a. Complete the square: x^2 + (b/a)x + (b/(2a))^2 = -c/a + b^2/(4a^2). Factor the left side: (x + b/(2a))^2 = (b^2 - 4ac)/(4a^2). Take the square root: x + b/(2a) = +/- sqrt(b^2 - 4ac)/(2a). Solve for x.

The discriminant D = b^2 - 4ac determines the nature of the roots: D > 0 gives two distinct real roots, D = 0 gives one repeated real root, D < 0 gives two complex conjugate roots.

Part II — Equation Solving

Linear Equations

A linear equation in one variable has the form ax + b = 0 (a != 0), with solution x = -b/a.

Strategy: Isolate the variable through inverse operations. Add/subtract to move terms, multiply/divide to clear coefficients. Preserve the equality by performing the same operation on both sides.

Quadratic Equations

Three methods in order of preference:

1. Factoring (when applicable): x^2 - 5x + 6 = 0 factors as (x-2)(x-3) = 0, giving x = 2 or x = 3.
2. Completing the square (always works): useful when the vertex form is also needed.
3. Quadratic formula (always works): the universal fallback.

Worked example (factoring). Solve 6x^2 + x - 2 = 0.

We seek two numbers whose product is 6(-2) = -12 and whose sum is 1. These are 4 and -3. Rewrite the middle term: 6x^2 + 4x - 3x - 2 = 0. Factor by grouping: 2x(3x + 2) - 1(3x + 2) = 0, so (2x - 1)(3x + 2) = 0. Therefore x = 1/2 or x = -2/3.

Polynomial Equations (Degree >= 3)

Rational Root Theorem. If the polynomial p(x) = a_n x^n + ... + a_0 has a rational root p/q in lowest terms, then p divides a_0 and q divides a_n. This gives a finite list of candidates to test.

Factor Theorem. r is a root of p(x) if and only if (x - r) divides p(x).

Strategy for cubics and quartics:

1. List rational root candidates via the Rational Root Theorem.
2. Test candidates using synthetic division or direct substitution.
3. Once a root r is found, divide by (x - r) to reduce the degree.
4. Repeat until the remaining factor is quadratic, then use the quadratic formula.

Worked example. Solve x^3 - 6x^2 + 11x - 6 = 0.

Rational root candidates: +/-1, +/-2, +/-3, +/-6. Test x = 1: 1 - 6 + 11 - 6 = 0. Yes. Divide by (x - 1) via synthetic division to get x^2 - 5x + 6 = (x - 2)(x - 3). Solutions: x = 1, 2, 3.

Part III — Systems of Equations

Substitution

Solve one equation for one variable, substitute into the other.

Worked example. Solve: x + y = 5 and 2x - y = 1.

From equation 1: y = 5 - x. Substitute into equation 2: 2x - (5 - x) = 1, so 3x = 6, x = 2, y = 3.

Elimination

Add or subtract multiples of equations to eliminate a variable.

Worked example. Solve: 3x + 2y = 7 and 5x - 2y = 9.

Add the equations: 8x = 16, so x = 2. Back-substitute: 6 + 2y = 7, y = 1/2.

Gaussian Elimination

For larger systems, reduce the augmented matrix to row echelon form using three row operations: swap rows, multiply a row by a nonzero scalar, add a multiple of one row to another.

Worked example. Solve: x + 2y + 3z = 9, 2x - y + z = 8, 3x + 0y - z = 3.

Augmented matrix: [1 2 3 | 9] [2 -1 1 | 8] [3 0 -1 | 3]

R2 <- R2 - 2R1: [0 -5 -5 | -10] R3 <- R3 - 3R1: [0 -6 -10 | -24]

R2 <- R2 / (-5): [0 1 1 | 2] R3 <- R3 + 6*R2: [0 0 -4 | -12]

R3 <- R3 / (-4): [0 0 1 | 3]

Back-substitute: z = 3, y + 3 = 2 so y = -1, x + 2(-1) + 3(3) = 9 so x = 2.

Solution: (x, y, z) = (2, -1, 3).

Gauss-Jordan variation. Continue to reduced row echelon form (RREF) so each pivot is 1 and all other entries in pivot columns are 0. This eliminates back-substitution.

Matrix Methods

A system Ax = b has solution x = A^(-1)b when A is invertible. For 2x2 matrices, the inverse has a closed form. For larger systems, use Gaussian elimination to find A^(-1) or apply Cramer's rule (determinant-based, practical only for small systems).

Cramer's Rule. For an n x n system Ax = b where det(A) != 0, x_i = det(A_i) / det(A), where A_i is A with column i replaced by b.

Part IV — Algebraic Structures

Groups

A group (G, *) is a set G with a binary operation * satisfying:

1. Closure: a * b is in G for all a, b in G.
2. Associativity: (a * b) * c = a * (b * c).
3. Identity: There exists e in G such that a * e = e * a = a for all a.
4. Inverses: For each a in G, there exists a^(-1) such that a * a^(-1) = a^(-1) * a = e.

If additionally a * b = b * a for all a, b, the group is abelian (commutative).

Examples. (Z, +) is an abelian group with identity 0. The symmetric group S_n (all permutations of n elements under composition) is non-abelian for n >= 3.

Rings

A ring (R, +, *) is a set with two operations where (R, +) is an abelian group, multiplication is associative, and multiplication distributes over addition. A ring with multiplicative identity 1 is a ring with unity. A ring where multiplication is commutative is a commutative ring.

Example. (Z, +, *) is a commutative ring with unity. The ring of n x n matrices over R is a ring with unity (the identity matrix) but is non-commutative for n >= 2.

Fields

A field (F, +, *) is a commutative ring with unity where every nonzero element has a multiplicative inverse.

Examples. Q, R, and C are fields. Z is NOT a field (2 has no multiplicative inverse in Z). The integers modulo p (Z/pZ) form a field when p is prime.

Why it matters for equation solving. The quadratic formula works over any field of characteristic != 2. Gaussian elimination works over any field. Understanding when you're working in a field vs. a ring determines which algebraic manipulations are valid.

Part V — Modular Arithmetic

Working modulo n means computing remainders upon division by n. Two integers a and b are congruent modulo n (written a = b mod n) if n divides (a - b).

Arithmetic rules. If a = b (mod n) and c = d (mod n), then:

• a + c = b + d (mod n)
• a * c = b * d (mod n)
• a - c = b - d (mod n)

Division requires care: a/c (mod n) exists only when gcd(c, n) = 1, in which case the modular inverse c^(-1) exists.

Worked example. Compute 7^100 mod 11.

By Fermat's Little Theorem, since 11 is prime and gcd(7, 11) = 1: 7^10 = 1 (mod 11). Therefore 7^100 = (7^10)^10 = 1^10 = 1 (mod 11).

Fermat's Little Theorem. If p is prime and gcd(a, p) = 1, then a^(p-1) = 1 (mod p).

Euler's Theorem (generalization). If gcd(a, n) = 1, then a^(phi(n)) = 1 (mod n), where phi(n) is Euler's totient function.

The Chinese Remainder Theorem. If gcd(m, n) = 1, then the system x = a (mod m), x = b (mod n) has a unique solution modulo mn. This decomposes modular problems into simpler components.

Part VI — Polynomial Theory

The Division Algorithm

For polynomials f(x) and g(x) with g != 0, there exist unique polynomials q(x) (quotient) and r(x) (remainder) such that f(x) = g(x) * q(x) + r(x) with deg(r) < deg(g) or r = 0.

The Fundamental Theorem of Algebra

Every non-constant polynomial with complex coefficients has at least one complex root. Consequently, every degree-n polynomial over C factors completely into n linear factors (counted with multiplicity).

This theorem (proved by Gauss in his doctoral dissertation, 1799) explains why the complex numbers are algebraically closed — no further extension is needed to solve polynomial equations.

Vieta's Formulas

For the polynomial a_n x^n + a_{n-1} x^{n-1} + ... + a_0 with roots r_1, ..., r_n:

• Sum of roots: r_1 + r_2 + ... + r_n = -a_{n-1}/a_n
• Sum of products of pairs: sum(r_i * r_j) = a_{n-2}/a_n
• Product of all roots: r_1 * r_2 * ... * r_n = (-1)^n * a_0/a_n

These relate coefficients to roots without solving the polynomial.

Part VII — Inequalities

Properties of Inequalities

• Adding the same quantity to both sides preserves direction.
• Multiplying by a positive number preserves direction.
• Multiplying by a negative number reverses direction. (The most common source of inequality errors.)
• Squaring both sides is valid only when both sides are non-negative.

Key Inequalities

AM-GM (Arithmetic Mean - Geometric Mean). For non-negative reals a_1, ..., a_n: (a_1 + a_2 + ... + a_n)/n >= (a_1 * a_2 * ... * a_n)^(1/n), with equality iff all a_i are equal.

Cauchy-Schwarz. For real numbers a_i, b_i: (sum a_i * b_i)^2 <= (sum a_i^2)(sum b_i^2), with equality iff a_i/b_i is constant.

Triangle Inequality. |a + b| <= |a| + |b|, with equality iff a and b have the same sign (or one is zero).

Worked example (AM-GM). Prove that for positive reals x and y, x/y + y/x >= 2.

Proof. By AM-GM applied to x/y and y/x: (x/y + y/x)/2 >= sqrt((x/y)(y/x)) = sqrt(1) = 1. Multiply by 2: x/y + y/x >= 2. Equality holds iff x/y = y/x, i.e., x = y.

When to Use This Skill

• Solving equations of any type (linear, quadratic, polynomial, systems)
• Simplifying or factoring algebraic expressions
• Working with modular arithmetic or number theory
• Proving algebraic identities or inequalities
• Analyzing algebraic structures (groups, rings, fields)

When NOT to Use This Skill

• For geometric reasoning — use geometric-intuition instead
• For calculus-based computation (limits, derivatives, integrals) — use numerical-analysis
• For sequence discovery or conjecture generation — use pattern-recognition
• For real-world problem formulation — use mathematical-modeling

Cross-References

• gauss agent: Systems of equations, Gaussian elimination, number theory applications. Primary agent for computational algebra.
• noether agent: Abstract algebraic structures, symmetry, ring and field theory. Named for Emmy Noether, whose work on abstract algebra and conservation laws transformed modern mathematics.
• proof-techniques skill: Proof methods used to establish algebraic results.
• pattern-recognition skill: Conjecture generation from algebraic patterns (e.g., discovering Vieta's formulas experimentally).
• numerical-analysis skill: Numerical root-finding when algebraic methods reach their limits.

References

• Artin, M. (2011). Algebra. 2nd edition. Pearson.
• Axler, S. (2015). Linear Algebra Done Right. 3rd edition. Springer.
• Gauss, C. F. (1799). Demonstratio nova theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. Doctoral dissertation.
• Hardy, G. H., & Wright, E. M. (2008). An Introduction to the Theory of Numbers. 6th edition. Oxford University Press.
• Lang, S. (2002). Algebra. 3rd edition. Springer GTM 211.
• Strang, G. (2016). Introduction to Linear Algebra. 5th edition. Wellesley-Cambridge Press.