Chapter 6

Polynomials

6.1 Adding Polynomials

6.1 Adding Polynomials

Monomial – one term expression Binomial – two term expression…. Polynomial – “many terms” What is a Term? What does “like terms” mean?

The degree of a term is the power of the variable in that term…

Determine the degree of the term:3x3x3xy

Determine the degree of the polynomial:3x+5x+27x+2x+1

Rule for Adding Polynomials:

Combine like terms! This means add or subtract the

numbers (called coefficients) in front of the variables…

Ex: 3x + 7x = 10x Ex: 5x + 6x² = 11x

Your Turn:

(6x² + 5x -7) + (5x +2)

(11xy-3y² - 4xy + 2) + (-6xy – 7xy + 4y² - 9)

HW 6.1 #13-50 odd

6.2 Subtracting Polynomials

Agenda

Warm-up 6.2 Subtracting Polynomials Practice subtracting 6.3 Multiplying Polynomials

Warm-up

Simplify 3x² + 2x – 6 - 5x² - 7x -3

Subtracting polynomials:

Distribute the negative sign..

Ex: (5x – 2) – (7x – 3)= 5x – 2 – 7x + 3

= -2x + 1

Your Turn:

(12x + 5) – (9x – 11)

(3x + 2x – 2) – (4x + 4x – 7)

HW 6.2 #1-43 odd

6.3 Multiplying Monomials

Multiplying Monomials

Remember, a monomial is a ONE term math expression

Every monomial is the product of factors

Ex: 6m²n = 2·3·m·m·n

Three Important Rules: Product of Powers:

Power of a power:

Power of a Product

nmnm xxx

nmnm xx )(

nnn yxyx )(

Product of Powers:

This is the idea that when multiplying polynomials, you add the exponents

Ex: x·x = x

Your turn: 3y·4y = ?

Power of a Power

When raising a polynomial to a power, multiply

Ex: (x)=x

Your Turn: (m)=?

Power of a Product When raising a product to a power,

distribute:

Ex: (3a)² = 3²·a² = 9a²

Your turn: (2pq)³ = ?

HW: 6.3 #1 – 43 odd

6.4 Multiplying a Polynomial by a Monomial

Warm-Up

(-x³y)²

(-2ab²)³(5a²b³)²

(3x)² - 7 + 2x² + 5

Multiplying a Polynomial by a Monomial:

Use the distributive property…

Ex. 1: 7x(5y + 7) = 7x·5y + 7x·7 = 35 xy + 49 x

Ex. 2: 4x²(2yz + 5z) = 4x²·2yz + 4x²·5z

= 8x²yz + 20x²z

Your Turn:

8m(9m² + 6m + 3)

2v³(12vp² - 7)

-7x²y(-3x – 7y – 12)

HW: 6.4 #1 – 31 odd

6.5 Multiplying Polynomials

The FOIL Method

FOIL stands for:

First – Outside – Inside – Last

You should get four terms when multiplying two binomials. Your answer may only have three terms if you combine the two like terms.

FOIL:

Ex.1: (x + 5)(x – 7)

= x·x + x·7 + 5·x + 5·7= x² +7x + 5x + 35= x² + 12x + 35

FOIL:

Ex. 2: (2x – 1)(x + 8)

= 2x·x + 2x·8 + (-1)·x + (-1)·8

= 2x² + 16x + (-1)x + (-8)= 2x² + 15x - 8

Your Turn:

(x + 3)(x + 2)

(x + 2)(x – 2)

(3x -5)²

HW: 6.5 #1 – 43 odd

Agenda

Warm-Up Homework Review 6.4 and 6.5 Practice Layers

* Adding/Subtracting* Multiplying Monomials* FOILing

Warm-Up

x³·x²

(x + 3)(x – 4)

(2x + 1)(x – 6)

6.6 Dividing Polynomials

Quotient Rules

Think of a polynomial as the product of its factors…

22

222

4

8

x

xxx

x

x

2

3

2

3 3

Example: Simplifying Quotients

233

4

35

2

3

4

33

223

9

12

11

y

x

yyyxxx

yxxxx

yx

yx

vvvvv

v

v

v

vvvvv

vv

v

v

Example: Power of a Quotient

25

4

5

2

5

2 2

2

222nnn

Divide a polynomial:

Divide each term of the numerator by the denominator:

3

24

3

24

3

24

33

3

53

23

52

23

233

6

10

6

18

6

1018

a

b

ab

bbaaaa

bbba

bbaaaa

baaa

ba

ab

ba

ba

ba

abba

HW: 6.6 #1 – 29 odd