8G Slides Ch 5 Algebraic Expressions.pptx

8G Chapter 5

5.2.1 - Monomials and

Binomials

• _{Key Skill: WWBAT use a geometric}

Key Vocabulary

mathematical expression – Examples: x, 5, 3x, xy, x2

• Remember, combining like terms

refers to adding (or subtracting) a bunch of terms

– Example: 3x2 + 2x + 7 - x2 - 3x -2 contains

Key Vocabulary

• Like terms meet both of two tests

1) They must have the same variable, AND

2) The variable must be raised to the same exponent

• 4x3 and 2x3 are like terms

• 3x and 3x2 are not like terms

Combining Like Terms

combining like terms:

Combining Like Terms

combining like terms:

k + 4k2 + 13 - 2k3 + 2k - 6 + 6k3 + 3k2

Distributive Property

Example

• How about this one:

Distributive Property

Example

• How about this one:

x(2x+4) + y(y-2) + 3x + 2x2 + 5y

Remember…..

• Multiply what’s on the outside of the

parenthesis by EVERYTHING on the inside:

Remember…..

• Multiply what’s on the outside of the

parenthesis by EVERYTHING on the inside:

How about Division?

• The distributive property applies here

as well. We must divide EVERYTHING by 2 just as we would multiply

EVERYTHING by 1/2.

More Examples

(8x3 + 16x2 + 4x + 24) ÷

4

More Examples

(8x3 + 16x2 + 4x + 24)

÷ 4

2x3 + 4x2 + x + 6

(4x3 + 8x2 + 6x) ÷ 2x

Factoring

5x + 25

Factoring

5x + 25

5 is the common factor Divide each term by 5

Answer: 5(x+5) 4x3 + 3x2 + x

x is the common factor Divide each term by x

What is the area?

x

What is the area?

• One way to see the area is as x(x+1)

x

What is the area?

• Another way: Area of square = x2 and area

of small rectangle is 1x, so total is x2 + x

x

How about this one?

p p

p

How about this one?

3p(2p+4) = 6p2 + 12p

p p

p

p p 1 1 1 1 p2 p2 p2 p2 p2 p2 p p

p p p p p

p p p p

5.2.2 - Multiplying Binomials

• _{Key Skill: WWBAT use a geometric}

Key Vocabulary

• _{Monomial is a variable expression with}

a single term: x, 7x, x3, 8x5, or 4xy

• _{Binomial is a variable expression with 2}

terms: x+1, 3x-5y, 4x2-7, 9xyz+4

• Trinomial is a variable expression with 3

Geometric Model

• What is the area of the large rectangle?

x

x

2

Geometric Model

• We can look at it as length times width

or (x+3)(x+2)

x

x

2

Geometric Model

• Or we can look at it in pieces that add

up to x2 + 5x + 6

Geometric Model

• Thus, a way to think about multiplying

(x+3)(x+2) is by using an “area” model and assigning one binomial to the

“length” and the other to the “width”.

• Note that multiplying two binomials

Another Example

• What is the area of the large rectangle?

x

x

3

Another Example

x2 + 8x + 15

x

x

3

5

x2 5x

5.2.5 - Multiplying Binomials

• _{Key Skill: WWBAT to use the}

Geometric Model

• To multiply (x+3)(x+2) we can use the

area model.

x

x

2

Geometric Model

• We can look at it in pieces that add up

to x2+5x+6

Using a Table to Multiply

to use a table to display the Distributive Property.

x 3

x

Using a Table to Multiply

to use a table to display the Distributive Property.

x 3

x x2 3x

Finishing the Problem

Another Example

2x 4

3x

Another Example

• Multiply: (2x+4)(3x-2)

• Add up the answers to get: 6x2 + 8x - 8

2x 4

3x 6x2 12x

Now You Try

Expand the following expression: (x + 6)(x + 8) =

(3x + 2)(5x + 9) =

Now You Try

Expand the following expression: (x + 6)(x + 8) = x2 + 14x +48

(3x + 2)(5x + 9) = 15x2 + 37x + 18

FOIL Method

• The quickest way of multiplying binomials is

to use the Distributive Property directly:

• (2x+3)(x+1)=

– Multiply the First term in each (2x · x)=2x2

– then the Outside terms (2x · 1)=2x – then the Inside terms (3 · x)=3x

– and finally the Last terms (3 · 1)=3

FOIL Method

multiplying a binomial by another binomial.

– For problems like: (x + 4)(x2 + 2x - 8)

Example

• Problems involving subtraction are

treated the same way, we must just be careful with signs.

• Find the product of the following:

Example

• (x + 5)(x - 4) can be treated like

(x + 5)(x + (-4))

• First: x times x = x2

• Outside: x times -4 = -4x • Inside: 5 times x = 5x

More Examples

More Examples

(x + 9)(x + 1) First: x2

Outside: 1x Inside: 9x Last: 9

Total x2 + 10x + 9

(4x - 4)(5x - 2) First: 20x2

Outside: -8x Inside: -20x Last: 8

Factoring

• Look at the trinomial: x2 + 7x + 6

• What were the two binomials that

were multiplied to get the trinomial?

Factoring

• Look at the trinomial: x2 + 7x + 6

• What were the two binomials that

were multiplied to get the trinomial?

Classwork

• Page 232, #1-8

• No need to draw rectangle diagrams,

5.3.1 - Special Products:

Perfect Squares

• _{Key Skill: WWBAT expand squared}

Example

Example

• Is it the same as 72 + 32 ? No!!!

• We know 102 = 100 and we know that

(7+3)2 = 100

• Is 72 + 32 the same thing? No!!!

Example

• First we rewrite as (x+2)(x+2)

• Then we can use the FOIL method to

More Examples

More Examples

(x+5)2

(x+5)(x+5) First: x2

Outside: 5x Inside: 5x Last: 25

Total: x2 + 10x + 25

(x+6)2

(x+6)(x+6) First: x2

Outside: 6x Inside: 6x Last: 36

Patterns?

• Do we see a pattern when we square a

Patterns?

• Do we see a pattern when we square a

binomial?

What about Subtraction?

What about Subtraction?

How do we solve: (x-5)2

Rewrite as: (x-5)(x-5) First: x2

Outside: -5x Inside: -5x Last: 25

Factoring

• Look at the trinomial: x2 + 16x + 64

• What was the binomial that we

squared to get the trinomial?

Factoring

• Look at the trinomial: x2 + 16x + 64

• What was the binomial that we

squared to get the trinomial?

5.3.2 - Special Products:

Difference of Squares

• _{Key Skill: WWBAT identify patterns in}

Differences of Squares

Differences of Squares

(x+2)(x-2) =

First: x2

Outside: -2x Inside: 2x Last: -4

More Examples

More Examples

(x+4)(x-4) First: x2

Outside: -4x Inside: 4x

Last: -16

Total: x2 - 16

(3x+1)(3x-1) First: 9x2

Outside: -3x Inside: 3x

Last: -1

Patterns

• Do we see any patterns from this kind

Patterns

• Do we see any patterns from this kind

of problem?

Mental Math

• How could we use this pattern to

Mental Math

• How could we use this pattern to

quickly multiply 54 by 46?

• (50+4)(50-4) = 502 - 42

• 2500 - 16 = 2484

• Note: this only works when the EXACT

Factoring

• What were the two binomials

multiplied together?

Factoring

• What were the two binomials

multiplied together?

Classwork

Multiplying Trinomials

• _{Key Skill: WWBAT use the distributive}

More Than Two Terms

multiply EVERYTHING inside one set of parenthesis by EVERYTHING inside

another set of parenthesis.

• How would we find the product for the

Example

• (x+1)(2x+y+3)=

• Add it up to get 2x2+5x+xy+y+3

x 1

2x 2x2 2x

y xy y

Examples

(3x - 1)(4x2 - 7x + 3) (4x2 + 5x - 9)(6x2 - x +

Examples

(3x - 1)(4x2 - 7x + 3)

12x3-21x2+9x

-4x2+7x-3

12x3-25x2+16x-3

(4x2 + 5x - 9)(6x2 - x +

7)

24x4 - 4x3 + 28x2

30x3 - 5x2 + 35x

-54x2 + 9x - 63

24x4 +26x3 -31x2 +44x

Classwork

1) (x+2)(x2+4x-5) =

2) (x-2)(2x2-8x-4) =

3) (x2+3x-1)(3x2-3x+1) =

Classwork

1) (x+2)(x2+4x-5) = x3 + 6x2 + 3x - 10

2) (x-2)(2x2-8x-4) = 2x3 – 12x2 + 12x + 8

3) (x2+3x-1)(3x2-3x+1) = 3x4 + 6x3 – 11x2 + 6x - 1

5.1.4 - Finding Angle Measures

• _{Key Skill: WWBAT write equations}

Find the Missing Angle

and solve the equation:

Find the Missing Angle

and solve the equation:

• 135 + x = 180, therefore x = 45º

Find the Missing Angles

and solve the equation:

58º

Find the Missing Angles

and solve the equation:

• 90 + 58 + x = 180, therefore x = 32º

58º

Find the Missing Angles

and solve the equation:

Find the Missing Angles

(4x-35) + (4x+35) = 360 8x = 360

x = 45

4x - 35 = 145º

4x + 35 = 215º 4x-35

Regular Polygon Angles

Regular Polygon Angles

Total Degrees _{Each Angle} 180° 60°

360° 90°

540° 108°

Formulas

• Total Degrees of the angles in a

Polygon = 180(n-2)

‘n’ is the number of sides in the polygon

• Degrees of one angle in a regular

polygon =

180(n - 2)