• Collect or combine the like terms:
• 3x – 6 + 2x – 8
1) 4x2 + 3x – 7 - 4x2 + 2x
2) 5x2y + 10xy - 2x2y + 5y – 6xy
5x - 145x - 14
5x - 75x - 7
3x3x22y + 4xy +5yy + 4xy +5y
• The first thing we will do with polynomials• is classify. There are two ways we can do• this:
– By the number of terms– By the degree
• Terms are the “chunks” of numbers and/or variables separated by the + or – signs.
• The degree of a polynomial is the largest exponent on a variable.
What is a TERM??
• The basic unit of algebra is a term.• 6a -xy 4b2 -0.7p
• A term has three (3) parts:
1. A sign, being positive(+) or negative (-).2. A number, called the coefficient.3. It may or may not contain a letter, called the variable.
What is a TERM??
• Every term must have a sign. If there isn’t one, assume there is an unwritten positive sign. Negative signs will always be written.
• 2x -4s 3x -9xy
+ 2x+ 3x
What is a TERM?
• Every term MUST have a number. If there isn’t one, assume the unwritten coefficient of one (1).
• What is the coefficient of each term?-x 2y -4xy x2
-1 2 -4 1
What is a TERM??
• Every term may or may NOT have a variable. If there is a variable and it has an exponent above the variable, 2x2, then the exponent belongs ONLY to the variable and not the whole term.
•
• EXAMPLES• 3x -x2 (4x)5 3xy2
Complete the chart below:
Example Sign (+ or -) Coefficient Variable(s)
8f8f
-10r-10r
4xy4xy
-22-22
-xyz-xyz
Classifying by terms…# of terms Classifying Word Example
1 Monomial
2* Binomial
3* Trinomial
4 or more* Polynomial*
x8−
42 −x
252 3 +− xx
12534 ++− xxxI suppose mathematicians lost interest in naming polynomials after 3 terms. Anything 4 terms or more gets the boring label of plain ol’ polynomial. Remember, terms are separated by a + or - sign, which then becomes a positive or negative for the term to follow!!
Polynomials
• A polynomial is an expression which consists of one or more terms.
• So…
– A monomial is a polynomial– A binomial is a polynomial– A trinomial is a polynomial
Classify each of the following according to the number of terms.
1. 3xy2. 2x + 3y - 6 3. x2 - 3y + 9 - 6x4. 4x2y2z4
5. 2x -56. 9
monomial
trinomial
polynomial
monomial
binomial
Constant; NOT a polynomial
Classifying by degree• To classify by degree you must know the
difference between a single variable term, such as 3x and a multi-variable term such as 3x2y2.
Example:
2z2 3xy3 -x2y3z4 5m6
Single variable
term
Multi variable
term
Multi variable
term
Single variable
term
single variable terms
• To determine the degree of a single variable polynomial, simply look for the term with the largest exponent.
Example:4x + 3x3 + 9y + 4 3rd degree7y2 + 3x - 9 2nd degree8x4 + 9y9 9th degree
multi-variable terms
• To determine the degree of a multi-variable polynomial, you must add all the exponents within a each individual term, then take the highest number.
• Example: 3x2y4 + 5xy8 + 5 9th degree
5xy2z5 + 3x3y4 8th degree
4 + 2 = 6 1 + 8 = 9
1 + 2 + 5 = 8 3 + 4 = 7
Try these:
1. 2x2 + 3x - 9
2. 4x2y4 + 2xy2 + 9
3. 5x - 16
4. 2x2y2 - 9x + 2y7
5. 7
6. X2 + 2x2y + 4y2
Degree = 2
Degree = 6
Degree = 1
Degree = 7
Degree = 0
Degree = 3
Classifying by degree…
Degree (largest exponent on a variable)
Classifying Word Example
0 Constant 9 (yep, a plain ol’ number)
1 Linear
2 Quadratic
3 Cubic
4 Quartic
25 10x −
4 37 32 2 1x x x− + + 38 23 2x x− + −
7 12x−
Polynomials with a degree higher than 4 are not named at this level !!
Your Turn…Classify by Term & Degree
• 1 Term – Monomial• Degree of 0 - Constant
78
Notice that classifying by terms has NOTHING to do with classifying by degree!
Does order matter?• Polynomials are usually arranged in one of two ways:
– Ascending order (smallest degree to largest degree)
– Descending order (largest degree to smallest degree)53 56 xxx ++
4 3 22 8 5 1x x x x+ − + +
When a polynomial is written in descending order the coefficient of the first term is called the leading coefficient.
Operations with Polynomials• Adding polynomials is simply combining like terms.
• (Like terms have the same exact variable and degree of exponents!)
• Example:
)297()523( 32234 xxxxxx −+++−1) List terms in descending order
3x4 - 2x3 + 9x3 + 5x2 + 7x2 - 2x
2) Add or subtract the coefficients of like terms:
3x4 + 7x3 + 12x2 - 2x
Example• Tip: Be careful with subtraction…watch your signs!
)572()1611( 22 +−−−+ xxxx11x2 - 2x2 + 6x + 7x - 5
9x2 + 13x = 5
Complete Adding and Subtracting Polynomials
Worksheet!!
Do your best on this worksheet and bring them back tomorrow!!