AP Precalculus - Topic 1.4, 1.5, 1.6

Degree

Largest exponent

Leading coefficient

coefficient of the variable with highest degree

Relative (Local) Extrema

A polynomial has a relative min or max where the function changes from increasing to decreasing or vice versa

Absolute (Global) Extrema

The greatest relative max (absolute max) and the lowest relative min (absolute min); doesn’t include other relative extremas; infinity and negative infinity are not considered an absolute max or min; if domain is restricted, endpoints are considered absolute extremas

What is the polynomial end behavior for a positive leading coefficient and even degree?

Ends of graph both go up

What is the polynomial end behavior of a negative leading coefficient and even degree?

Ends of graph both go down

What is the polynomial end behavior of a positive leading coefficient and an odd degree?

Left end goes down and right end goes up

What is the polynomial end behavior of a negative leading coefficient and an odd degree?

Left end goes up and right end goes down

What is the domain for all polynomials?

All real numbers

What is the range for odd degree polynomials?

All real numbers

What is the range for even degree polynomials?

The range is restricted; a calculator is needed to find the “floor” or “ceiling” of the function

How do you find the x-intercepts of a polynomial?

x-intercepts = degree (n)

How do you find the max # of relative extremas of a polynomial?

One less than degree (n-1)

How do you find the # of points of inflection of a polynomial?

Two less than degree (n-2)

How do you find the y-intercepts of a polynomial?

Always one (all functions have one y-intercept); f(0) = constant

Difference of two perfect squares

A² - B² = (A + B)(A - B); any even exponent is considered a perfect square

Multi-Stage Factoring

factoring problems, like graphs, that require multiple steps, have no constant, factor out GCF

Difference of two perfect cubes

Take cube root of boths terms; set first parenthesis (binomial) as (A + or - B); the second one (trinomial) should start with A² , then A*B, and lastly B²; use “SOAP” (Same, Opposite, Always Positive) to get the proper signs

Grouping

Will always be a cubic and have four terms; first split the four terms into groups of two; factor each smaller group by factoring out a GCF; what’s inside the parentheses should be the same & terms outside will combine into one parenthesis

Non x² trinomials

Factoring with these trinomials will follow classic order (x + constant)(x+ constant), but replace x’s with x² ‘s

Complex Zero Theorem

A polynomial degree of (n) is guarenteed to have exactly (n) “complex” zeros (when taking into account multiplicity); complex #: A + Bi (both imaginary and real #’s

Multiplicity of Zeros

depending on the power of a factor, it dictates how the function behaves at its x-intercept and what the graph will look like at that x-interept; n = 1 will slice through x-axis; n = 2 will bounce off the x-axis; n = 3 will squiggle through the x-axis

Is the degree equal to the type of difference (1st, 2nd, 3rd, etc.)?

Yes, it’s equal to the type of difference when it’s constant; e.g. if a function has a degree of 3 (cubic), then it will have a constant 3rd difference