Polynomial BasicsAdding and Subtracting

Polynomials

MM1A2 a

• 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!!

What is NOT a polynomial term???

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

C

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

• 4 Terms – Polynomial• Degree of 3 - Cubic

13524 23 +−+− xxx

Your Turn…Classify by Term & Degree

• 2 Terms – Binomial• Degree of 1 - Linear

30 36x−

Your Turn…Classify by Term & Degree

• 3 Terms – Trinomial• Degree of 2 - Quadratic

29 4 5x x− −

Your Turn…Classify by Term & Degree

• 2 Terms – Binomial• Degree of 4 - Quartic

4 8x x−

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.

Essential Question…

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

4

More Practice

Match the polynomial to the correct degree!!

1.

2n2 + 6n - 8

D.

2.

D

3.

(2r3 + 12r2 - r) + (3r3 + 7r - 6)

A

4.

A

5.

C

6.

A

7.

B

Essential Question…

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!!

Multiplying Polynomials!!

Operations with Polynomials

• Multiplication is basically distribution.

)632(5 234 +− xxx

Another Example

)67)(12( +− xx

Another Example

3( 2)(4 3)x x+ −

Another Example

2(4 1)( 6)x x x+ − +