Algebra: Polynomial, Roots, and Factoring Key Concepts

Binomial

A polynomial with two terms (e.g., x² + 3x)

Monomial

A number, variable, or product of both with whole-number exponents (e.g., 5x²).

Trinomial

A polynomial with three terms (e.g., 2x² + 5x + 2)

Polynomial

A monomial or sum of monomials

Degree of a Monomial

Sum of the exponents of all variables (e.g., 3x²y¹ has degree 3)

Degree of a Polynomial

Greatest degree among its terms

Leading Coefficient

Coefficient of the first term in standard form

Standard Form of a Polynomial

Terms written in order of decreasing exponents.

Factored Form

A polynomial written as a product of its factors

Factored Completely

A polynomial written as a product of unfactorable polynomials with integer coefficients

Factoring by Grouping

Using the distributive property to factor a four-term polynomial.

FOIL Method

A shortcut for multiplying two binomials (First, Outer, Inner, Last)

Zero-Product Property

If the product of two numbers is 0, at least one is 0

Roots

The solutions of a polynomial equation.

Repeated Roots

Roots that appear more than once

Parabola

The U-shaped graph of a quadratic function

Axis of Symmetry

The vertical line that divides a parabola into two symmetric parts

Vertex

The lowest point (if it opens up) or highest point (if it opens down) on a parabola

Vertex of a Parabola

The turning point where the function changes from increasing to decreasing

Vertex Form

f(x) = a(x - h)² + k

Intercept Form

f(x) = a(x - p)(x - q).

Average Rate of Change

Change in y over change in x: (f(b) - f(a)) / (b - a)

Zero of a Function

The x-value where f(x) = 0 (x-intercept)

Maximum Value

The y-coordinate of the vertex when the parabola opens downward (a < 0).

Minimum Value

The y-coordinate of the vertex when the parabola opens upward (a > 0).

Even Function

f(-x) = f(x) for all x (symmetric about y-axis)

Odd Function

f(-x) = -f(x) for all x (symmetric about origin)

Quadratic Equation

Equation in the form ax² + bx + c = 0, where a ≠ 0

Quadratic Function

Function in the form y = ax² + bx + c, where a ≠ 0

Completing the Square

Adding a constant to make x² + bx + c a perfect square trinomial

Discriminant

Expression b² - 4ac from the quadratic formula (tells number/type of roots)

Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a).

Conjugates

Binomials like a + b and a - b.

Counterexample

Example that disproves a statement.

Radical Expression

An expression containing a radical (e.g., √50).

Like Radicals

Radicals with the same index and radicand (e.g., 3√5, 2√5)

Simplest Form (of a Radical)

No perfect powers under the radical, no radicals in denominators

Rationalizing the Denominator

Removing radicals from the denominator by multiplying by a form of 1

System of Nonlinear Equations

A system where at least one equation is nonlinear (e.g., includes x² or y²)