Algebra 2 Cheat Sheet

The core ideas of Algebra 2 distilled into a single, scannable reference — perfect for review or quick lookup.

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Quick Reference

Polynomial Division

Polynomial division rewrites an improper rational expression as a quotient plus a remainder term over the divisor. Long division works for any polynomial divisor, while synthetic division is a shortcut for divisors of the form x-c.

Remainder and Factor Theorems

When p(x) is divided by x-c, the remainder is p(c). If p(c)=0, then x-c is a factor. These theorems connect substitution, factoring, and zeros.

Rational Expression Simplification

Rational expressions simplify by factoring and canceling common nonzero factors. Domain restrictions from the original denominator remain after cancellation.

Holes and Vertical Asymptotes

Canceled denominator factors create holes (removable discontinuities). Non-canceled denominator zeros create vertical asymptotes.

Horizontal and Slant Asymptotes

Degree comparison determines end-behavior asymptotes for rational functions. Equal degrees give a horizontal asymptote at leading-coefficient ratio; numerator degree one higher gives a slant asymptote from division.

Crossing a Horizontal Asymptote

A horizontal asymptote describes long-run behavior, not a forbidden y-value. A rational graph may cross it at finite x-values.

Complex Numbers and i

Complex numbers extend the real system with i, where i^2=-1. They allow solutions to equations with negative discriminants.

Conjugate Root Structure

If a polynomial with real coefficients has root a+bi, then a-bi must also be a root. This preserves real coefficients in expanded form.

Key Terms at a Glance

Rational Function:A function of the form p(x)/q(x) where p and q are polynomials and q(x)!=0.

Hole:A removable discontinuity caused by a canceled denominator factor.

Vertical Asymptote:A line x=a that the graph approaches with unbounded values near a.

Horizontal Asymptote:A line y=L approached as x goes to positive or negative infinity.

Slant Asymptote:A linear asymptote found when numerator degree is one greater than denominator degree.

Remainder Theorem:Remainder from dividing p(x) by x-c equals p(c).

Factor Theorem:x-c is a factor of p(x) iff p(c)=0.

Synthetic Division:A coefficient-based shortcut for division by x-c.

Complex Number:A number in the form a+bi with i^2=-1.

Conjugate:For a+bi, the conjugate is a-bi.

Discriminant:For ax^2+bx+c, b^2-4ac determines root type.

Extraneous Solution:A computed value that does not satisfy the original equation or domain.

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