Algebra II Unit 1

polynomial

an algebraic expression consisting of one or more terms with:

only addition, subtraction, and multiplication

only non-negative integer exponents

monomial

one term expression

binomial

two term expression

trinomial

three term expression

standard form of a polynomial

arranges term in order from greatest to least exponents

degree of a polynomial

value of the highest(or sum) variable exponent

coefficient

number attached to a variable by multiplication

leading coefficient

number attached to the variable with the highest degree

constant term

number that does not contain a variable

conjugates

a pair of binomials with opposite operations

function

a relationship in which each input has exactly one output

identity

an equality which is true for all of the variables

what is the goal of verifying identities?

show that the left side equals the right side

when proving an identity,

do not move terms from side to side

steps to verify identities on a calculator with only one letter variables

1. enter the left side of the equation into y1

2. enter the right side of the equation into y2

3. check the table to see if the outputs are the same

steps to verify identities on a calculator when multiple variables are used

1. plug in a random value for each variable

2. find total for left side

3. find total for right side

4. see if they match

the cost function

C(x) = R(x) - P(x)/R(x) = C(x) + P(x)/P(x) = R(x) + C(x)

composition of functions

when one function is inside another function(works from right to left)

vertical line test

used to determine if a graph represents a function

one to one function

each x has only one y and each y has only one x

inverse function

a reflection over the line y = x

to graph an inverse function,

switch the x and y values

to write an inverse equation,

you switch the x and y values then solve for y

inverse notation

f^-1(x)

to verify whether two functions are inverses equations,

you must show that both these statements are true:

f(f^-1(x)) = x

f^-1(f(x))

factor

a number or quantity that divides another number or quantity, leaving now remainder

greatest common factor(gcf)

the greatest factor that goes into two or more expressions evenly

perfect square

a number or variable that can be expressed as the product of two equal factors

factoring two perfect squares can only be done when…

the terms are separated by a minus sign

the resulting product of factoring the difference of two perfect squares are…

conjugate pairs

when factoring by grouping, the polynomial must contain…

at least four terms

perfect square trinomials identity

a²+2ab+b²=(a+b)(a+b) or (a+b)²

a²-2ab+b²=(a-b)(a-b) or (a-b)²

unlike factoring perfect square binomials, perfect cube binomials…

can be factored when separated by a plus or minus sign

variables are a perfect cube when…

the exponent is a multiple of three

steps of factoring by u-substitution

1. recognize that the constant term does not have a common binomial factor and therefore, factoring by grouping is not an option

2. write a let statement to establish that the binomial expression will be replaced with a “u” and substitute “u” in the expression for everywhere you see the binomial

3. factor the polynomial expression in terms of “u”

4. substitute the binomial back into the expression for “u” and simplify

zero product property

if a x b = 0 then either a = 0, b = 0, or both equal zero

x-intercepts

where the graph crosses the x-axis(must be real)

zeros/roots and solutions

values of x that make f(x) = 0(can be imaginary)

substitution method

a solution to a system of equations is the location(x, y) where the graphs intersect

steps of the substitution method

1. isolate a variable in one equation

2. substitute the isolated expression into the second equation for that variable

3. simplify and solve for the first variable

4. plug your answer into an original equation to find the second variable

elimination method

1. rewrite both the equations so the common terms are lined up

2. multiple one or both equations to generate opposite values in one variable

3. add the equations and solve the result for the first variable

4. substitute your solution into an original equation to find the second variable