polynomials

x^2 - 2xy+ y^2

(x-y) (x+y)

graphing polynomial functions

- zeros are what touches the x intercepts - multiplicty is the number of times that a factor appears - if there is a simple zero or any zero with an odd multiplicty, its graph crosses the x axis - zeros follow the exponents in the equation

guide to factoring polynomials

1. ask are there any perfect squares or cubes and the amount of terms that factor GCF - two terms diff. of squares sum of cubes diff. of cubes gcf - three terms the british method perfect square trinomials

solving rational equations

1. cross mutiply and then equal equations to itself 2. solve for the variable (equal to 0) 3. equal to 0 then get x itself if possible or if its a trinomial or quadratic equal to 0 OR 1. mutiply both sides by a common den. - determine the excluded values that can equal 0 - combine expressions together 2. simplify

multiplying rational expressions

1. factor out, factor out first, factor anything common out of all hte variables and apply the rules 2. simplify 3. then foil or cross mutiply, try to cancel out

simplify rational expressions

1. find GCF 2. cancel out the common if possible SIGNS MATTER

adding and subtracting rational expressions

1. find the LCD 2. multiply both denominator and numerator by the LCM 3. make sure the denomniator is the same 4. add or subtract the numerator through combining like terms 5. simplify- by keeping it in factors and then canceling out any common ones

subtracting polynomials

1. first reverse the sign of each term in the 2nd polynomial 2. group like terms together 3. combine like terms

dividing rational expressions

1. keep change flip factor out first, factor anything common out of all hte variables and apply the rules

dividing a polynomial by a monomial

1. seperate the denominator and each term on its own 2. divide each of those 3. combine like terms

zeros of the polynomial function

1. set function to 0 2. find GCF to all coefficents and variables 3. divide out GCF 4. set each factor to zero, even GCF

factoring trinomial and quadratics where a doesnt equal 1

1. set to 0 2. find 2 integers that whose product equals ac and sum equals b 3. replace the coefficent b of the middle term bx in with the sum of integers and distribute the variable 4. then do steps 3 and 4 above PLEASE MAKE SURE TO COMPLETE THE WHOLE EQUATION

factoring using grouping

1. set to zero 2. collect terms into expression of two groups 3. factor GCF out of each group where you should have an identical minomial factor in each term 4. set each factor to 0

synthesis division polynomial

1. set your divisior equal to zero ex: x-1 =0 divisor - x=1 2. write out all coefficents above the graph and line it up in the correct form 3. take the 1st coefficent and put it under the line then multiply it by the divisor then you put that under the 2nd coefficent then add and repeat over 4. once you went through all of the coefficents put x^4, x^3, x^2, etc. next to the coefficents, the last # would be a constant and the # in the box is the remainder - you have to go one down for the exponent when labeling

a double zero

A double zero occurs when one of the factors of the polynomial is squared.

multiplying polynomials

FOIL it tells you to multiply the first terms in each of terms except each other mutiply outer terms and closest terms combine like terms

what are distinct x intercepts

The distinct xxx‑intercepts are -2−2minus, 2, -1−1minus, 1, 111, and 222. The word distinct means that we do not list any of them twice.

when given a polynomial and then ask to find the rest of the zeros after the other zeros were given you...

We know that if we multiply all of the binomial factors, the final term will equal the product of the second term (including sign) of each factor. use zero x zero x varaiable of unknown zero = constant of the equation solve for the variable

odd polynomial function

all degrees are odd

polynomial

an expression comprised of variables, exponents and coefficents

factoring trinomial and quadratics where a = 1

ax^2 + bx +c 1. set to zero 2. find two integers whose product equals c and whole sums up to b 3. form a product of factors containing the variable and each of the integers found 4. set each factor equal to 0 and solve to find zeros

when matching up polynomials to a graph

decide the number of zeros by the highest degree whether is postive or negative then find the zeros of the actual equation to prove

difference of squares

find the perfect square even for the exponents, doesnt have to have an exponent plug it into an equation simplify,

long division polynomials

from the book

adding polynomials

like terms 1. place like terms together 2. add those like terms

to determine the number of distinct zeros

look at the highest degree and how many (x-#) represents one zero

when mutiplying 3 factors

mutiply the first two and then the third one last

when asked to present all zeros you have to ignore the first number

so 3(x-#) ignore 3 3 also does not equal 0

even polynomial function

symmetry across x axis all degrees are even

to simplify polynomials

you have to first combine like terms

evaluating a polynomial expression

you plug constant into the variables and solve for the equation x^3 + 5x^2 + 1, x=1 (1)^3 + 5 (1)^2+ 1= 5