Polynomial Functions

methods to solve quadratic functions

- by factoring, zero prod property, only real solutions - use quadratic formula, real and non-real solutions - by graphing, find zeros, x intercepts - complete the square

Names of polynomials using degree

0 - constant, 1 - linear, 2 - quadratic, 3 - cubic, 4 - quartic, 5 - quintic

Names of polynomials using number of terms

1 - monomial or polynomial or 1 term 2 - binomial or polynomial of 2 terms 3 - trinomial or polynomial of 3 terms 4 - polynomial of 4 terms

how to find roots of 3rd degree or higher polynomial (with a graphing calculator)

1. fundamental theorem of alg will tell you # of roots 2. use graph calc to id any real roots 3. factor out the factors using synthetic division 4. solve resulting quadratic using easiest method

rational root theorem

If P(x) is a polynomial function with integer coefficients, the only possible rational roots: factor or constant term ------------------------------------ factor of leading coefficient *creates a finite list of possible rational roots of a polynomial equation with rational coefficients

conjugate root theorem

If a polynomial P(x) with rational coefficients, then irrational roots that have the form a + √b occur in conjugate pairs. a − √b is also a root If a polynomial P(x) with real coefficients, then complex roots occur in conjugate pairs. if a + bi is a complex root , a − bi is also a root

polynomial function of x

a polynomial with the variable x

monomial

a real number, a variable, or the product of these

multiplicity

a root that appears k times has a multiplicity of k ie. p(x) = (x − 2)²(x + 4) 2 is a zero with multiplicity of 2 −4 is a zero with multiplicity of 1

Pascal's Triangle

a triangular array of numbers in which the first and the last number of each row is 1. each of other numbers is sum of the 2 numbers above it row #0.........1 row #1.........1 1 row #2.........1 2 1 row #3.........1 3 3 1 row #4.........1 4 6 4 1 row #5..........1 5 10 10 5 1

standard form of polynomial function

arranges the terms by degree in descending numerical order ex. P(x)=4x³(cubic term)+3x²(quadratic term)+5x(linear term)-2(constant term)

number of turning points for a polynomial function with a degree n (n ≥ 1) is generally....

at most n − 1 turning points

The fundamental Theorem of Algebra

every polynomial equation of degree n ≥ 1 -has exactly n roots including multiple and complex roots -has n linear roots -has at least one complex zeros

degree of a monomial (in one variable)

exponent of the variable

first step in factoring

factor out the GCF, if there is one

zero product property

if the product of 2 (or more) factors is zero, then one or both (or more than one of the) factors must equal zero.

how does multiplicity affect the graph

multiplicity of 1 will be close to linear multiplicity of 2 will be close to quadratic multiplicity of 3 will be close to cubic and so on

turning points of polynomial function

places where the graph changes direction. the degree determines the maximum number of these.

Factor Theorem

the expression x − a is a factor of a polynomial only if "a" is a zero of the polynomial function

degree of polynomial (in one variable)

the greatest degree of its monomial terms

x intercept of a polynomial function

the point(s) where the graph crosses the x axis. This is also called the root(s) or zero(s)

relative maximum and relative minimum

when graph of a polynomial function has several turning points: relative max is value of function at up to down turning point relative min is value of function at down to up turning point

Steps to factoring

1. factor out GCF, if there is one 2. if 4 terms, consider factor by grouping 3. check of special cases - difference of 2 squares - difference or sum of 2 cubes 4. quadratic trinomial, find factors x game

how to find roots of 3rd degree or higher polynomial (without a graphing calculator)

1. fundamental theorem of alg will tell you # of roots 2. identify possible roots (rational root theorem) 3. identify true root(s) using synthetic division 4. solve resulting quadratic using easiest method

synthetic division

A shorthand method of dividing by a linear binomial of the form (x-a) by writing only the coefficients of the polynomials.

polynomial

a monomial or a sum or difference of monomials.