polynomials
how to check when using long division
(divisor) (quotient) + remainder = dividend
linear divisor
(x^1 - c)
origin symmetry
f(-x)= -f(x) - opposites
axis symmetry
f(-x)= f(x)
number of turning points
n-1
extrema
places where the graph changes directions (peaks, valleys) - AKA relative min and relative max
polynomial equation theorem #5
sum of roots product of roots
if you have a linear divisor
synthetic division is possible
f(c) will equal
the remainder
fundamental theory of algebra
whatever the degree is, is how many answers you'll have
test symmetry
- axis symmetry - origin symmetry
polynomials
- the highest exponent is first - ends in a constant
sum of roots
-An-1/An top A is the opposite of the second coefficient (b) bottom A is the leading coefficient (A) - B/A
characteristics of polynomials
1. exponents are whole numbers (no negative exponents) 2. there are no variables in the denominator 3. no variables contained within absolute value or square root symbols 4. they are smooth (has rounded curves) and continuos (ongoing) and must have a degree of 2 or higher
how to work backwards with polynomials
1. find the zeroes 2. find the multiplicity of each zero 3. find the number of turning points 4. find the degree 5. write function 6. find y intercepts
leading coefficient test
1. if A>0 and the degree is ODD, the graph will start at the bottom and move to the top 2. if A>0 and the degree is EVEN, both ends of the graph will go up 3. if A<0 and the degree is ODD, the graph will start at the top and move to the bottom 4. if A<0 and the degree is EVEN, both ends of the graph will go down
Properties of Roots of Polynomial Equations
1. if a polynomial equation is of degree n, then counting multiple roots separately, the equation has n roots 2. if a+bi is a root of a polynomial equation with real coefficients , (b doesn't equal zero) then the imaginary number a-bi is also a root. Imaginary roots, if they exist, occur in conjugate pairs
there are 2 ways to find the remainder
1. synthetic division 2. plug in p(c)
product of roots
Ao/An (if n is even) -Ao/An (if n is odd) Ao or -Ao is constant (C) An is the leading coefficient (A) C/A or -C/A
polynomial long division
a method used to divide polynomials
polynomial equation theorem #2
complex conjugate theorem - if p(x) is a polynomial...if a+bi is a root then a-bi is also a root
multiplicity
even (graph touches and turns around at the x axis) odd (graph goes through the x axis)
possible rational answers can be found by doing
factors of p/factors of q
graphing polynomials
find the end behavior using the leading coefficient test
intermediate value theorem
finds out if there is an existence of a real zero (an x intercept) - if f(a) and f(b) have opposite signs, there is at least one value "c" between "a" and "b" for which f(c)=0 that is a zero of the graph
polynomial equation theorem #3
if a+b{i (square root of i) is a root then a-b{i is also a root
rational zero (root) theorem
if f(x) is a polynomial of integer coefficients and p/q in lowest terms is a rational zero of f(x), then p is a factor of the constant and q is a factor of the leading coefficient
polynomial equation theorem #1
if p(x) is a polynomial of degree "n", then the equation p(x)=0 has exactly "n" roots
polynomial equation theorem #4
if p(x) is of an odd degree, then p(x)=0 has at least one real root
when the remainder is zero, whatever (x-c) is,
is a factor of f(x)
