Problem-solving recipes

A step-by-step procedure for every problem type this course teaches — what to do first, what to do next, and how to tell which case you're in.

Fundamentals of algebra

Factor a polynomial completely

Use when: you're handed a polynomial and asked to write it as a product of factors, none of which can be split any further over the integers

Goal: the polynomial's complete factorization, with every factor checked and re-checked until nothing more comes apart

Polynomials and factoring

Equations & inequalities

Solve an absolute value equation or inequality

Use when: an absolute value expression sits somewhere in an equation or inequality

Goal: every valid solution to an equation, or the complete solution set to an inequality, with impossible cases ruled out before any needless case-splitting

Absolute value and inequalities

Solve a polynomial inequality

Use when: a polynomial is compared to 0 (or can be rearranged to be) and you need every x making it positive, negative, or zero

Goal: the complete solution set in interval notation, with every boundary correctly included or excluded

Polynomial inequalities · extended later in this course

Graphs & functions

Find the inverse of a function

Use when: you have a formula for a one-to-one function f (or one already restricted to a piece where it's one-to-one) and need a formula for f⁻¹

Goal: a verified formula for f⁻¹(x), confirmed to undo f in both directions, with its domain stated whenever it isn't all real numbers

Composition and inverse functions

Polynomial & rational functions

Solve a polynomial equation using the Rational Zero Theorem

Use when: you're given a polynomial equation of degree 3 or higher, with integer coefficients, that doesn't factor by inspection

Goal: every real zero of the polynomial, found exactly, with no rounding

The Rational Zero Theorem

Graph a rational function

Use when: you're given $f(x) = p(x)/q(x)$ and asked to sketch or fully describe its graph

Goal: a complete graph: every hole, every asymptote, both intercepts, and the correct sign in every region

Asymptotes, holes, and graphs of rational functions

Solve a rational inequality

Use when: you're given an inequality like $p(x)/q(x) > 0$ (or $<, \ge, \le$ ), a fraction of two polynomials

Goal: the exact solution set in interval notation, with every hole and vertical asymptote correctly excluded

Asymptotes, holes, and graphs of rational functions

Exponential & logarithmic functions

Solve an exponential equation

Use when: the unknown sits in an exponent, anywhere in the equation

Goal: every exact value of the unknown, checked against the original equation

Solving exponential and logarithmic equations

Solve a logarithmic equation

Use when: the unknown sits inside one or more logarithms

Goal: every exact value of the unknown, verified against the domain of the original equation

Solving exponential and logarithmic equations

Trigonometric functions

Solve a right triangle

Use when: you're given a right triangle with two known pieces of information beyond the free 90° angle (two sides, or one side and one acute angle) and need every side length and angle measure

Goal: all three side lengths and both acute angle measures

Solving right triangles

Graphs of trigonometric functions

Graph a sinusoidal function from its equation

Use when: you're given $y=A\sin(B(x-C))+D$ or $y=A\cos(B(x-C))+D$ (or an equation that can be rewritten into that shape) and asked to sketch or describe its graph

Goal: the midline, amplitude, period, and phase shift, plus enough plotted points to sketch one full cycle

Amplitude, period, and phase shift

Trigonometric identities & equations

Verify a trig identity

Use when: you're given two trig expressions joined by an equals sign and asked to show they're the same for every $\theta$ in the domain, not asked to solve for a specific $\theta$

Goal: one side rebuilt, one legal rewrite at a time, into something literally identical to the untouched other side

Simplifying and verifying trigonometric identities

Solve a trig equation that needs an identity first

Use when: a trig equation has more than one function name, or the same function at two different powers, so the reference-angle method from the previous section doesn't apply directly

Goal: every solution in the given interval (or the general solution), found only after the equation has been rebuilt into single-function form

Equations requiring identities

Applications of trigonometry

Choose the right law and solve an oblique triangle

Use when: you're given three pieces of a triangle (some mix of angles and sides), no right angle is guaranteed, and you need to find the rest

Goal: every angle and side of the triangle, using the correct law from the start instead of guessing and backtracking

Law of Cosines: SAS and SSS triangles

Find the nth roots of a complex number

Use when: you need every complex number w with w^n = z, not just the single real root a calculator's root button would return

Goal: all n roots, each with the correct modulus and angle, evenly spaced around one circle

nth roots of complex numbers

Analytic geometry

Analyze a conic from general form

Use when: you're handed a second-degree equation $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ and asked to identify the conic and pin down where it sits

Goal: the conic's family, plus its center, vertices, foci, or asymptotes read straight off a standard-form equation

Identifying a conic from its equation · extended later in this course

Rotate away the cross term (Step 2 continued, when B ≠ 0)

Use when: Step 2 of the general-form recipe found $B\neq0$ , so the conic is rotated and needs its axes turned before Steps 3 and 4 can read the equation

Goal: a new equation in $x'$ and $y'$ , with no $x'y'$ term, ready to hand back to Step 3

Rotation of axes

Complete the square when the leading coefficients aren't 1 (Step 4 continued)

Use when: Step 4 of the general-form recipe needs to complete the square, but $A$ and/or $C$ aren't both 1

Goal: a fully factored, standard-form equation, ready for Step 5's center, vertex, and focus read-off

Translated conics and completing the square

Systems, matrices & inequalities

Solve a system by Gaussian elimination and back-substitution

Use when: you have two or more linear equations in the same variables and need every value that satisfies all of them at once

Goal: the augmented matrix in row echelon form, then the exact solution (or a parameterized description of infinitely many solutions, or a certification that none exist) read straight off it

Matrices and row reduction

Decompose a rational expression into partial fractions

Use when: you have a proper rational expression $p(x)/q(x)$ whose denominator factors into distinct linear factors

Goal: the equivalent sum of simple fractions, one term per factor, with every numerator constant pinned down

Distinct linear factors · extended later in this course

Decompose a rational expression with a repeated linear factor

Use when: the denominator's factorization includes a linear factor raised to a power greater than 1, like $(x-r)^k$ with $k \ge 2$

Goal: one numerator term for every power from 1 up to $k$ of the repeated factor, plus the usual single term for every distinct factor, with every constant pinned down

Repeated linear factors · extended later in this course

Decompose any rational expression into partial fractions

Use when: the denominator factors into any mix of distinct linear, repeated linear, and irreducible quadratic factors (repeated or not)

Goal: the complete term-by-term decomposition, with every constant solved for, whatever the denominator's factorization looks like

Irreducible quadratic factors

Sequences, series & probability

Prove a statement by mathematical induction

Use when: you're asked to prove a statement $P(n)$ true for every integer $n$ from some fixed starting point $n_0$ onward, and testing every case individually is impossible

Goal: a complete proof that $P(n)$ holds for every integer $n \ge n_0$ , resting on exactly two checked facts

The induction principle