Definition: Two or more numbers that can not be reduced to a single sum, as a result of variables.
Etymology: Poly-nomials means many-numbers like poly-gon means many-sides.
Polynomials are numbers containing variables separated by + or – signs. A constant has a constant value that does not change, 2 always equals 2, as is 2 = 2. Constants like 0, 1, 2 or 3 and variables like x, y, z are monomials. Monomials have one term. A binomial has two terms. If a quantity can be simplified to a single term then it is a monomial, if it cannot then it is a polynomial. Binomial is “two monomials separated by a + or – â€.
The “2 + x†is called a binomial and the 17 is a constant. A regular number is a constant and a constant or variable is a monomial. A constant has the same value constantly no matter the equation and a variable has different values that vary from equation to equation. A variable varies from number to number and a constant does not.
constant = 1 a constant is just a number
variable = x a variable is a letter in place of an unknown number
mixed term= 2x a mixed term has a constant in front of a variable.
binomial= 2x +1 two terms separated by operators + or –.
trinomial = x2 +x + 5 two terms separated by operators + or –.
polynomial with four terms = x3 + x2 +x + 5 four term polynomial
polynomial with five terms = x4 + x2 + 2x +x + 5 five term polynomial
I have a cookie I dropped it, oh no!
It broke into three pieces.
How much is each piece?
I don't really know...
When we don't know we use a variable. When we have to add variables that can not be a single number because we write it like “piece 1 = piece 2 + piece 3†and then get fancier and use letters “piece A + piece B + piece Câ€.
We have three pieces and say a+b+c = 1 cookie. Each piece is some fraction that is a mystery for now.
17 – 2 = ? A subtraction problem
17 – 2 = x Replace a blank spot with a variable, x is convenient
15 = x Simplify
15 = 15 Missing number is 15 “a=a†is the reflexive property
When we start we need to find what x equals, we call this solving for x. We see x = 15. x on the left side and 15 is on the right side of the equation by itself. This is called x is isolated. When x is isolated on one side and the other side has a single term we call this x is solved.
Basic Structure of a Polynomial
2x monomial (Greek for one number)
x^2 + 2x binomial, (Greek for two numbers)
x^2 + 2x + 1 trinomial (Greek for three numbers)
6^x3 + 5x^2 -1x + 6 This has four monomials so it is a four term polynomial
Constant: 5 or another defined immutable quantity.
Variable: x or another letter or symbol used for unknown values.
Term: A monomial separated from another by an operation sign.
Coefficient: The constant factor in a monomial.
The degree of a polynomial: The exponent of the leading term.
The terms of a polynomial are the individual numbers with or without a variable. A mixed term is a number with a coefficient and a variable, a constant term is a number without a variable. We say degree to specify whether we are dealing with a value, vector, quadratic, cubic, quaternion, quintion, or higher polynomial. Each of those polynomials have an x variable raised to a power corresponding with the order listed. A value is a number having degree 0, vector is almost a linear equation having 1°, a quadratic is 2°, a cubic 3° and so on.
A monomial is any single term not having a + or – separately it from another term. 2, 3, x, 2x, 3x^4 are all monomials. Adding two monomials that are unlike creates a binomial because they can not simplify into a single term. A binomial takes the form of x+n such as x+1. If however they are like then you get a larger monomial.
2 is a monomial because it has one term. 2+x is a binomial because it has two terms. 2 + x can not be simplified any further so it stands as its own number or value until we can solve for x by substituting a value for it. A binomial is a number of two different sums acting as one since we can not solve for it and get a single value.
If we take 2 + x and do 2 + x + x we get 2x + 2 since x + x is 2x. If we take 2x + 2 and do 2x + 2 + 4 we get 2x + 6. Polynomials are the result of sums that could not simplify any further. They are single terms having an unknown variable separated only by ± symbols while being multiplied by the constant number in front of them, which is called their coefficient. Any arithmetic operation can be performed on them but might change their status as a polynomial to something else. We are only interested in polynomials and rational expressions for algebra. A polynomial might have a radical operation on it and still be considered a polynomial if criteria are met.
If a polynomial contains a negative power then it really contains a square root and is called a signomial
If a polynomial contains a fractional exponent then it is called a posinomial.
If a polynomial contains a division of some x/some x then it is called a rational expression.
These math statements can be functions but are not considered true but rather pseudo polynomials.
√x^2 +2x – 3 is not a polynomial: posinomial
x^1/2 +2x – 3 is not a polynomial: posinomial
x^2 +2x-1 – 3 is not a polynomial: signomial
x+1/x – 1 is not a polynomial: rational expression
x^3+x^2–x–1/x^2+2x+1bbis a polynomial because it divides into a binomial with no fractional remainders.
x+1/x–1 has fractional “x term†remainders so it is a rational expression not a polynomial.
Polynomial Operations
Polynomials can be calculated using all 6 arithmetic operation + – ∙ / ^√.
Polynomial addition
You are at lunch and a friend is selling her dessert. You and another friend really want it.
Before anyone offers a price you say on of the:
(a) “I will match anyone's price plus $1...†or
(b) “I will double any offer...†or
(c) “I will double any offer plus $1...â€
You do not know what price your friend will say so how can we state what price you just agreed to pay? Unknown price+$1= x+1. This is written as “x+1 dollarsâ€. This is used in business and construction. You are going to pay x+1 (in dollars) what is your friend willing to pay. We could make a table of all or some possible guesses.
“Friends offerâ€
“Our offerâ€
$0.50
$1.50
$0.75
$1.75
$1
$2
$1.35
$2.35
Whatever we think he might be willing to pay we have to be ready to pay that plus another dollar.
Let the friends offer be “x†for “unknown†and our offer be “x+1â€:
If x = 0.5 then x3+13x2+50x+56 ÷ (x+7) 0.5+1 = 1.5
If x = 0.75 then 0.75+1 = 1.75
If x = 1 then 1+1 = 2
If x = 1.35 then 1.35+1 = 2.35
Our original equation was x+1. Replacing x with a constant is called substitution or substituting for x. If his number is x and our number is 1 adding them is x+1. Because x is a letter and 1 is a constant they can not combine to make one number. To combine we must substitute another number for x. By themselves x and 1 are single terms meaning single numbers so we call them monomials. “x†is a monomial and “1†is a monomial. When we add them we have a two term sum called a binomial. They cannot be simplified further so x+1 is the only way we can write it. We can add numbers like this and do other operations.
Exercise: Fill in the rest of the tables for offer b and c:
x
x + $1.00
$0.50
$1.50
$0.75
$1.75
$1
$2
$1.35
$2.35
x
2x
$0.50
2($0.50) = $1.00
$0.75
2($0.75) = $1.50
$1
2($1.00) = $2.00
$1.35
2($1.35) = $2.70
x
2x+1
$0.50
2($0.50) +1 = $2.00
$0.75
2($0.75) +1 = $2.50
$1
2($1.00) +1 = $3.00
$1.35
2($1.35) +1 = $3.70
Polynomial multiplication
We offered x + 1 when a third person shows up and says the y will pay $0.25 more than us. They are offering
(our price) + 0.25. We substitute x+1 into the grouping symbols:
(our price) + 0.25 = their offer Take data and form an equation with words.
(x+1)+0.25 = their offer Replace words with variables.
x+(1+0.25) = their offer Simplify by adding the constants together.
x+1.25 = their offer x + 1.25 is simplest form.
A fourth person shows up and says the will pay double the highest bidder: 2(last offer) = current offer
2(last offer) Replace words with what it asks you to; last offer was x+1.25
2(x+1.25) Multiply factor with every term inside ()
(2∙x+2∙1.25) We are multiply every term with the factor “2â€
(2∙x)+(2∙1.25) Using distributive property
2x+2.5 Simplified
Recap: 2(last offer) 2(x+1.25) (2x)+(2.5) 2x+2.5 is final answer
2(last offer) what we heard so wrote it down
2(x+1.25) substituted words with “the priceâ€
(2x)+(2.5) distributed the terms out of () and multiplied
2x+2.5 normally this is where we add the two term but can't because of the variable
2x+2.5 final answer
This shows how to add and multiply a simple real life word problem by taking data and converting it into an equation with words then substituting those words with numbers and variables to create a formula. A formula is an equation containing variables that explains how to get answer to something. Later we call these formulas functions. We used association to regroup () during addition and used distribution to regroup during multiplication.
Multiplying binomials to get a trinomial:
(x + 1) (x -1) Take two monomials
(x + 1) (x -1) first: (x) · (x) = x^2
(x + 1) (x -1) second: (x) · (-1)= -1x
(x + 1) (x -1) third: (1) · (x) = 1x
(x + 1) (x -1) fourth: (1) · (-1)= -1
x^2 -1x + 1x -1 add the terms together
x^2 + 0x -1 the “ x1 †terms cancel out
x^2 -1 we are left with a binomial.
This process is called F.O.I.L. Which details the process first, inside, outside, last. This is the recommended order to multiply terms but not strictly necessary. Later we will not write “0x†but leave it blank. It is offered for illustration so that you know you can fill gaps in a polynomial if you need to but likely won't.
The square of (x+1) is (x+1)^2. This means we are multiplying (x+1) · (x+1).
Anytime we square a number we multiply it by itself.
This also means that √(x+1)^2 = x+1. Because there is no fraction of x it is still a polynomial and not a posinomial.
x^2 + 2x + 1 has three terms: x^2, 2x, 1. A term is a quantity separated by plus or minus signs.
(x+1)^2 A binomial raised to a power of 2
(x+1) · (x+1) Written out in factor notation
x^2 +2x +1 The result after multiplying the factors
x^2 is a variable because it is a letter with no coefficient other than 1 which cancels itself out.
2x is a mixed variable because it has a constant.
1 is the constant because it is a numeral and not a variable.
Multiplying binomials to get a trinomial:
(x + 1) (x -1) Take two monomials
(x + 1) (x -1) first: (x) · (x) = x^2
(x + 1) (x -1) second: (x) · (-1)= -1x
(x + 1) (x -1) third: (1) · (x) = 1x
(x + 1) (x -1) fourth: (1) · (-1)= -1
x^2 -1x + 1x -1 add the terms together
x^2 + 0x -1 the “ x1 †terms cancel out
x^2 -1 we are left with a binomial.
This process is called F.O.I.L. Which details the process first, inside, outside, last. This is the recommended order to multiply terms but not strictly necessary. Later we will not write “0x†but leave it blank. It is offered for illustration so that you know you can fill gaps in a polynomial if y ou need to but likely won't.
The square of (x+1) is (x+1)2. This means we are multiplying (x+1) · (x+1).
Anytime we square a number we multiply it by itself.
This also means that √(x+1)^2 = x+1. Because there is no fraction of x it is still a polynomial and not a posinomial.
x^2 + 2x + 1 has three terms: x^2, 2x, 1. A term is a quantity separated by plus or minus signs.
(x+1)^2 A binomial raised to a power of 2
(x+1) · (x+1) Written out in factor notation
x^2 +2x +1 The result after multiplying the factors
x^2 is a variable because it is a letter with no coefficient other than 1 which cancels itself out.
2x is a mixed variable because it has a constant.
1 is the constant because it is a numeral and not a variable.
