Warm UpIntroduction to Polynomials and

Adding and Subtracting Polynomials

Classifying Polynomials:

We classify polynomials based on the number of _________.

What is a monomial?What is a monomial?

What is a binomial?What is a binomial?

What is a trinomial?What is a trinomial?

When do we use the term polynomial?When do we use the term polynomial?

terms

A polynomial with only 1 term.

A polynomial with 2 terms.

A polynomial with 3 terms.

We use the term polynomial to describe any expression which contains some combination of variables and numbers.

What is a term?

Terms are made up of numbers, variables or the product and/or quotient of some combination of numbers and variables.

EXAMPLE: 2x2x is a term. (the product of a # and variable)

2x/32x/3 is a term. (the quotient of a # and variable)

Terms are separated by ________ and Terms are separated by ________ and ________ signs!________ signs!

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Classify each of the polynomials

1. 3x2y + 3x + 4y

2. 2x2y4z

3. 2xy2 + 3x2y + 9x

4. 5x2y + 2xy3

Trinomial

Monomial

Trinomial

Binomial

Calculating the degree of a polynomial..

If it is a single variable term, find the term with the largest exponent.

5x3 + 9x2 + x

This is a 3rd degree polynomial.Remember the understood 1!!!

1

If it is a multi-variable term, you add the If it is a multi-variable term, you add the exponents of all of the variables.exponents of all of the variables.

4x2y + 2xy3 + 9

This is a 4th degree polynomial (3+1=4). Don’t forget the understood 1’s!!!!

1 1

Calculate the degree of the polynomial…

1. 6x2 + 4x + 4x3 + x

2. 21x2y + 12x4y2 + 2xy4

3. 3x2 + 2y3

3rd degree3rd degree

6th degree6th degree

3rd degree3rd degree

Writing a polynomial in standard form…

• Standard form is in descending order of the x variable.

1. 2xy3 + 3x2y + x3

2. 3xy2 + 3x3 + 2x2yX3 + 3x2y + 2xy3

3x3 + 2x2y + 3xy2

Adding and Subtracting Polynomials…

is nothing more than combining like terms. Remember you can do this vertically or horizontally.

1. (2x2 - 3x + 3) + (4x2 + 5x - 9)

2. (3x2 - 9x - 5) - (-2x2 - 4x + 5)6x6x22 + 2x -6 + 2x -6

5x5x22 - 5x - 10 - 5x - 10

5(x5(x22 - x - 2) - x - 2)

Ex 1 (x2 + 3x + 4) + (-2x2 + 10x - 5)

Combine LIKE terms

x2 + 3x + 4

-2x2 + 10x - 5

-1x2 + 13x - 1 Final Answer

Ex 2. (4b3 - 2b) + (b3 + 6b2 + 3b - 7)

4b3 + 0b2 - 2b + 0

b3 + 6b2 + 3b - 7

5b3 + 6b2 + 1b - 7 Final Answer

Ex 3. (12y2 - 8y + 4) - (9y2 + 5y + 1)

Don’t forget to Distribute the -1!!

(12y2 - 8y + 4) - 1(9y2 + 5y + 1)

(12y2 - 8y + 4) - 9y2 - 5y - 1

12y2 - 8y + 4

- 9y2 - 5y - 1

3y2 - 13y + 3 Final Answer

Ex 4. (3a3 + 10a - 15) - (-a3 + 2a2 + 6a - 9)

(3a3 + 10a - 15) - 1(-a3 + 2a2 + 6a - 9)

3a3 + 10a - 15 + a3 - 2a2 - 6a + 9

3a3 + 0a2 + 10a - 15

+a3 - 2a2 - 6a + 9

4a3 - 2a2 + 4a - 6 Final Answer

POLYNOMIAL MULTIPLICATION (2x - 3)(x + 5)

3 methods

a. Distribute (2x - 3)(x + 5) *4 multiplications

2x2 + 10x - 3x - 15 (then combine like terms)

2x2 + 10x - 3x - 152x2 + 7x - 15 Final Answer

b. 3rd grade style

2x - 3

x + 5

+10x - 15

2x2 - 3x + 0

2x2 + 7x - 15 Final

Answer!

c. Box Method

2x - 3

x 2x2 -3x

+ 5 +10x -15

2x2 + 7x - 15 Final Answer

Ex 5. (4x + 7)(3x + 7)

a.) 12x2 + 28x + 21x + 49

12x2 + 49x + 49

b.) 4x + 7 c.) 4x + 7

3x + 7 3x 12x2 +21x

+28x + 49 + 7 +28x + 49

12x2 + 21x + 0

12x2 + 49x + 49

Ex 6. (x - 1)(x2 - 4x + 6)

• x3 - 4x2 + 6x - 1x2 + 4x - 6x3 - 5x2 + 10x - 6

b. x2 - 4x + 6 c. x2 - 4x + 6

x - 1 x x3 -4x2 +6x -1x2 + 4x - 6 -1 -1x2 +4x - 6

1x3 -4x2 + 6x + 0 x3 - 5x2 + 10x - 6 x3 - 5x2 + 10x - 6

Ex 7. (2z2 + 3z - 4)(4z + 5)

• 8z3 + 10z2 + 12z2 + 15z - 16z - 208z3 + 22z2 - 1z - 20

b. 2z2 + 3z - 4 c. 2z2 + 3z - 4 4z + 5 4z 8z3 +12z2 -16z

+10z2 + 15z - 20 +5 10z2 +15z - 20 8z3 +12z2 - 16z + 0 8z3 + 22z2 - 1z - 20

Ex 8. (2a + 5)(2a - 5)

a. 4a2 - 10a + 10a - 25

4a2 - 25

b. 2a + 5 c. 2a +5

2a - 5 2a 4a2 +10a

-10a - 25 -5 -10a -25

4a2 +10a + 0

4a2 - 25 4a2 - 25

Ex 9. (3m + 4n)2

a. (3m + 4n)(3m + 4n)

9m2 + 12mn + 12mn + 16n2

9m2 + 24mn + 16n2

b. 3m + 4n c. 3m +4n

3m + 4n 3m 9m2 +12mn

+12mn + 16n2 +4n +12mn +16n2

9m2 + 12mn + 0

9m2 + 24mn + 16n2 9m2 + 24mn + 16n2