Algebra and Function Properties: Roots, Domains, and Polynomial Factoring

What determines the shape of x^n?

Whether n is positive/negative and even/odd.

What is the domain of x^n when n is positive?

All real numbers.

What is the range of x^n when n is positive and even?

[0, ∞)

What is the range of x^n when n is positive and odd?

All real numbers.

What is the domain of x^n when n is negative?

R − {0}.

What is the range of x^n when n is negative and even?

(0, ∞).

What is the range of x^n when n is negative and odd?

R − {0}.

What is the domain of √x?

[0, ∞).

What is the range of √x?

[0, ∞).

What are the domain and range of ∛x?

Both are all real numbers.

What is an n-th root function?

f(x) = x^(1/n) = ⁿ√x.

What is the domain of an even-root function?

[0, ∞).

What is the domain of an odd-root function?

All real numbers.

What is a root of a polynomial?

A value a where f(a) = 0.

What is the maximum number of distinct roots of degree n?

At most n distinct roots.

What does a root a imply about factoring?

x − a is a factor of f(x).

When is a a root of f(x)?

Exactly when f(x) = (x − a)g(x).

What is the polynomial division form?

f(x) = g(x)q(x) + r(x), with deg r < deg g.

What are q(x) and r(x)?

The quotient and remainder.

When does dividing by x − a give remainder 0?

Exactly when a is a root.

What is the Integer Root Theorem?

An integer root must divide the constant term.

What candidates should you test for integer roots?

The positive and negative factors of the constant term.

Does the Integer Root Theorem find non-integer roots?

No; it only restricts possible integer roots.

How do you find all polynomial roots?

Factor, then find the roots of each factor.

What is root multiplicity?

The exponent of its repeated factor (x − a)^m.

What is a simple root?

A root with multiplicity 1.

What is a double root?

A root with multiplicity 2.

What does (x − a)^3 mean for root a?

a has multiplicity 3.

What is the difference of squares formula?

a² − b² = (a − b)(a + b).

What is the difference of cubes formula?

a³ − b³ = (a − b)(a² + ab + b²).

What is the difference of fourth powers formula?

a⁴ − b⁴ = (a − b)(a³ + a²b + ab² + b³).

What is the general difference of powers formula?

a^n − b^n = (a − b)(a^(n−1) + ... + b^(n−1)).

What is the substitution factoring trick?

Write f(x) = g(x^m), factor g, then substitute x^m back.

What is the first step in solving a polynomial equation?

Move all terms to one side so the equation equals 0.

What is the second step in solving a polynomial equation?

Factor the polynomial as much as possible.

What is the final step in solving a polynomial equation?

Set each factor equal to zero and solve.

When are two functions equal?

Same domain and same output for every x in that domain.

What must equal polynomials have?

Same degree and identical coefficients.

What is the domain of p(x)/q(x)?

All real x where q(x) ≠ 0.

How do you find a rational function's domain?

Set the denominator equal to zero; exclude those x-values.

When is p(x)/q(x) = 0?

p(x) = 0 AND q(x) ≠ 0.

Can a denominator zero be a zero of a rational function?

No; it is excluded from the domain.

What happens when a common factor cancels?

The excluded x-value creates a hole.

What is a hole in a rational graph?

A missing point caused by a canceled factor.

What usually causes a vertical asymptote?

A denominator zero that does not cancel.

Why aren't canceled rational functions technically equal?

They can have different domains.