Glossary

Binomial Coefficient (n choose k)

Criticality: 2

The number of ways to choose k items from a set of n distinct items, represented as $\binom{n}{k}$, which serves as the numerical factor for each term in a binomial expansion.

Example:

The binomial coefficient $\binom{5}{2}$ is 10, meaning there are 10 ways to choose 2 items from a set of 5.

Binomial Theorem

Criticality: 2

A formula that provides a systematic way to expand expressions of the form $(a+b)^n$ for any positive integer n, using binomial coefficients.

Example:

Using the Binomial Theorem, $(x+y)^3$ expands to $x^3 + 3x^2y + 3xy^2 + y^3$ .

Degree (of a polynomial)

Criticality: 3

The highest power of the variable in a polynomial when written in standard form.

Example:

The polynomial $3x^4 - 2x + 1$ has a degree of 4.

Domain

Criticality: 3

The set of all possible input values (x-values) for which a function is defined, avoiding undefined operations like division by zero or square roots of negative numbers.

Example:

The domain of f(x) = 1/(x-3) is all real numbers except x=3, because division by zero is undefined.

End Behavior

Criticality: 3

The behavior of a function's graph as the input variable (x) approaches positive or negative infinity.

Example:

For $f(x) = -x^3 + 2x$ , the end behavior is that as x approaches positive infinity, f(x) approaches negative infinity, and as x approaches negative infinity, f(x) approaches positive infinity.

Factored Form

Criticality: 3

A polynomial or rational function expressed as a product of its linear or irreducible quadratic factors.

Example:

The function f(x) = (x-2)(x+3) is in factored form, clearly showing x-intercepts at x=2 and x=-3.

Holes (Removable Discontinuities)

Criticality: 2

Points where a rational function is undefined due to a common factor in the numerator and denominator that cancels out, resulting in a single point discontinuity.

Example:

The function h(x) = (x^2-4)/(x-2) has a hole at x=2 because (x-2) is a common factor that cancels out, leading to a point discontinuity.

Horizontal Asymptote

Criticality: 3

A horizontal line that the graph of a rational function approaches as x approaches positive or negative infinity, determined by comparing the degrees of the numerator and denominator.

Example:

The function $y = (2x+1)/(x-3)$ has a horizontal asymptote at y=2, determined by the ratio of leading coefficients.

Leading Coefficient

Criticality: 3

The coefficient of the term with the highest degree in a polynomial when written in standard form.

Example:

In the polynomial $4x^5 - 7x^2 + 1$ , the leading coefficient is 4.

Pascal's Triangle

Criticality: 2

A triangular array of numbers where each number is the sum of the two numbers directly above it, used to determine the coefficients in binomial expansions.

Example:

The third row of Pascal's Triangle (starting from row 0) is 1, 3, 3, 1, which are the coefficients for $(a+b)^3$ .

Polynomial Long Division

Criticality: 3

A systematic method for dividing one polynomial by another, similar to numerical long division, used to simplify rational expressions or find slant asymptotes.

Example:

Dividing $(x^2 + 5x + 6)$ by $(x+2)$ using polynomial long division yields a quotient of $(x+3)$ with no remainder.

Quotient (Polynomial Long Division)

Criticality: 2

The result of a polynomial long division, representing the whole part of the division, which can form the equation of a slant asymptote.

Example:

When dividing $x^2+3x+2$ by $x+1$ , the quotient is $x+2$ .

Range

Criticality: 2

The set of all possible output values (y-values) that a function can produce.

Example:

The range of the quadratic function f(x) = x^2 is [0, ∞), as its y-values are always non-negative.

Real Zeros

Criticality: 3

The x-values where a function's graph crosses or touches the x-axis, making the function's output zero. Also known as x-intercepts.

Example:

For f(x) = (x-1)(x+2), the real zeros are x=1 and x=-2, where the graph intersects the x-axis.

Remainder (Polynomial Long Division)

Criticality: 2

The polynomial left over after polynomial long division, whose degree is always less than the divisor.

Example:

When dividing $x^2+1$ by $x-1$ , the remainder is 2, meaning $x-1$ is not a factor.

Slant Asymptote (Oblique Asymptote)

Criticality: 2

A diagonal line that the graph of a rational function approaches when the degree of the numerator is exactly one greater than the degree of the denominator.

Example:

For $f(x) = (x^2+1)/(x-1)$ , performing polynomial long division reveals a slant asymptote at y = x+1.

Standard Form (Polynomial)

Criticality: 3

A polynomial written with its terms in descending order of powers of the variable, from the highest degree term to the constant term.

Example:

The polynomial $P(x) = 5x^4 - 2x^2 + 7x - 1$ is in standard form, with the highest power first.

Standard Form (Rational Function)

Criticality: 3

A rational function where both the numerator and denominator are polynomials expressed in standard form.

Example:

The rational function $R(x) = (3x^2 + 2x - 1) / (x^2 - 5)$ is in standard form, allowing easy identification of its horizontal asymptote.

Vertical Asymptotes

Criticality: 3

Vertical lines that the graph of a rational function approaches but never touches, occurring at x-values where the denominator is zero and the numerator is non-zero.

Example:

The rational function g(x) = (x+1)/(x-2) has a vertical asymptote at x=2, as the denominator becomes zero there.