UNIT 2 – POWERS AND POLYNOMIALS

Date Lesson Text TOPIC Homework

Feb.

20 2.1 3.3

Working With Exponents Pg. 127 # 1 - 7

WS 2.1

Feb.

21 2.2 3.3

Working with Exponents (Variables) Pg. 127 # 8

EHB Pg. 55 # 6, 7

Feb.

22 2.3 3.3

Multiplying/Dividing & Power Rule

with Monomials

Pg. 127 # 9, 10b, 14b,

EHB Pg. 55 # 8

Feb.

23 2.4 many

Like Terms with Albegra Tiles WS 2.4

Feb.

26 2.5 3.3

Like Terms without Algebra Tiles

QUIZ (2.1 – 2.3)

Pg. 134 # 1 – 6

,Pg. 151 # 1 - 9, 11

Feb.

27 2.6 3.5

Like Terms & Monomials

Evaluating

WS 2.6

Feb.

28 2.7 3.7

Distributive Property Pg. 166 # 1, 4 - 8

Mar. 1 2.8 3.6

More Expanding Simplifying and

Evaluating

QUIZ (2.4 – 2.6)

WS 2.8

Pg. 167 # 9, 15

Mar. 2 2.9 Algebra - Perimeter and Area WS 2.9

Mar. 5 2.10 Unit 2 Review

WS 2.10

Pg. 174 # 3, 4, 7 - 20

Mar. 7 2.11 UNIT 2 TEST

MPM1D Lesson 2.1 Working with Exponents

Law Example

Proof

(2 )(2 ) 2m n m n (3

2)(3

4) = 3

2 + 4 = 3

6

2 3 5(3 )(3 ) (3 3)(3 3 3) 3

22

2

mm n

n

66 2 4

2

33 3

3

3

1

2

3 3 3 33 3

3 33

(2 ) 2m n m n (3

2)3 = 2 33 = 3

6

2 3 2 2 2 6(3 ) (3 )(3 )(3 ) (3 3)(3 3)(3 3) 3

Ex. 1 Write in expanded form and then simplify:

a) 34 99 b) 25 99 c) 3

5

4

4

d) 5

6

8

8 e) 342 f) 427

Ex. 2 Write as a single power (where possible), then evaluate:

a) 53 22 b) 42 )3()3( c) 13

15

8

8

d) 42

8

44

44

e) 323 f) 24

2

g) 94

2512

3

33 h)

23

4

3

i)

423

85

23

232

Pg. 127 # 1 – 7

& WS 2.1

MPM1D Lesson 2.2 Working with Exponents (Variables)

You do it! Complete the table below in your groups.

Product/Quotient Expanded Form Single Power

4 3x x x x x x x x x

7 2w w

5 2y y y y y y y

y y

6 2m m

=

2 3 4 5c d c d c c d d d c c c c d d d d d

435 62 qpqp

c c c c c c c d d d

c c c c d d

5 3

3 2

16

4

k m

k m

2

4x x x x x x x x x

3

22 j m

1. Simplify:

a) 34 xx b)

37 bb c) yy 28 5

c) 4523 ba d) 42

1234

8

32

cab

cba

e) 223 5 yxyx

f) )8()4( 434 baba g) 2

225

8

)4()2(

ab

abba h)

)3(8

)2()3( 23424

yx

xyyx

Pg. 127 # 8

EHB Pg. 55 # 6, 7

MPM1D Lesson 2.3 Multiplying/Dividing and Power Rule with Monomials

Ex. 1 Simplify:

a) 34 99 b)

34 55 c) 34 yy

d) 4

6

x

x e) baba 362 f)

zyx

zyx25

326

Ex. 2 Simplify:

a) 2xy b) 342 yx c) 43z

Remember, when you multiply Remember, when you divide powers with the same base, powers with the same base,

you add the exponents. you subtract the exponents.

4 3 4 3 7x x x x

77 4 3

4

22 2

2

d) k

k

8

16 3

e) 524 23 xx f)

2

234

3

23

a

bba

Ex. 3 If 2x then evaluate the following expression:

32

7

2

48

x

x

When you multiply or divide terms that contain both coefficients and variable,

work with the coefficients and variables separately.

5 3 5 3

5 3

8

3 2 3 2

6

6

m m m m

m

m

6 66 3 3

3 3

8 84 4

22

a aa a

a a

Pg. 127 # 9, 10b, 14b,

EHB Pg. 55 # 8

MPM1D Lesson 2.4 Algebra Tiles

To organize a collection of algebra tiles, we group like tiles.

There are two x2 - tiles, three x - tiles, and five 1 - tiles.

These tiles represent the expression 2x2 + 3x + 5.

When a collection contains red and blue tiles, we group like tiles and remove zero pairs.

One –x2 - tile, three x - tiles, and two 1 - tiles are left.

We write –1x 2 + 3x + 2. We could also write –x

2 + 3x + 2.

The expression –x 2 + 3x + 2 has 3 terms: –x

2, 3x, and 2

Terms are numbers, variables, or the products of numbers and variables.

Terms that are represented by like tiles are called like terms.

– x2 and 3x2

are like terms. Each term is modelled with x2 - tiles. In each term, the variable x is

raised to the exponent 2.

– x2 and 3x are unlike terms. Each term is modelled with different sized tiles.

Each term has the variable x, but the exponents are different.

An expression is simplified when all like terms are combined, and any zero pairs are removed.

– x2 + 3x2 simplifies to 2x2

.

– x2 + 3x cannot be simplified.

Ex. Which expression does each group of tiles represent?

a) b)

c) d)

Ex. Identify terms that are like 3x:

–5x, 3x2, 3, 4x, –11, 9x

2, –3x, 7x, x

3

Ex. Identify terms that are like –2x2:

2x, –3x2, 4, –2x, x

2, –2, 5, 3x

2

Ex. Combine like terms. Write the simplified expression.

Ex. Simplify each expression. Use algebra tiles if you wish.

a) xx 365 b) 362 22 xx

c) xx 235 d) xxxx 323 33

e) xx 564 f) 725534 22 xxxx

Ex. 6 Simplify 42132 22 xxxx using algebra tiles. positive negative

WS 2.4

MPM1D Lesson 2.5 Like Terms without Algebra Tiles - Evaluating

YOU DO IT! Do # 1- 4 in your groups. Refer to the textbook pages 130 to 133 and fill in the blanks:

Weekly Shopping List:

1. For the expression yxyx 26 23 ,

TERM COEFFICIENT VARIABLE PART

2.

POLYNOMIAL CLASSIFICATION DEGREE

16 y

135 2 xx

237 qp

yxyx 26 23

3. Simplify the following expression: 2 4 3 3 2 6a b c a c a b c (Collect like terms first.)

4. Evaluate the above expression where 0.5, 0.75, 2.5a b c .

5. A jar is filled with yellow, green and red jellybeans. In a game you receive 5 points for each yellow

jellybean you pick and 2 points for every green jellybean you pick. You lose 3 points for every red jellybean

you pick.

a) Write a polynomial expression (including opening statements) for the number of points you receive.

Let y represent the number of yellow beans, g the number of green beans and r the number of red beans.

b) Determine the number of points you receive for 3 yellows, 1 green, and 2 reds.

6. Write two like terms for a) 7x b) 43y

7. Simplify by collecting like terms:

a) 6 3 4 4 2 7a p m p a m a b) 4 7 10 14 3q q

c) 2 23 5 2 1 9x x x d) 2 23 5 2 1 4y y y y

By now, you should be familiar with variables and exponents, and you may have dealt with expressions like 3x4 or 6x. Polynomials are sums of these "variables and exponents" expressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables (letters) which are raised to whole-number exponents (or else the terms are just plain numbers); The exponent is considered to be part of the variable. There are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions. Here are some examples:

26x This is NOT

a polynomial term...

...because the variable has a negative

exponent.

2

1

x

This is NOT a polynomial term...

...because the variable is in the denominator.

x This is NOT

a polynomial term... ...because the variable

is inside a root sign.

24x This IS a polynomial

term... ...because it obeys all

the rules.

Here is a typical polynomial:

When a term contains both a number and a variable part, the number part is called the "coefficient". The coefficient on the leading term is called the "leading" coefficient.

a one-term polynomial, such as 2x or 4x2, may also be called a "monomial"

("mono" meaning "one")

a two-term polynomial, such as 2x + y or x2 – 4, may also be called a "binomial"

("bi" meaning "two")

a three-term polynomial, such as 2x + y + z or x4 + 4x

2 – 4, may also be called a "trinomial"

("tri" meaning "three")

Pg. 134 # 1 – 6

Pg. 151 # 1 – 9, 11

MPM1D Lesson 2.6 Like Terms & Monomials - Evaluating

Ex. 1 Simplify and evaluate each of the following where 3x and 4y .

a) 3 2 4 5 5x y x y x b) 4 2 6 2x y x x y

c)

32

2

2

(3 )

x

y d)

3

2

3 4

2

x xy

xy

e) 2 24 2 ( )x y x y f) 2 2 2 2(2 ) (3 ) 3 9x y x y

WS 2.6

MPM1D Lesson 2.7 The Distributive Property

We use the distributive property when we multiply a constant by a polynomial.

We can model 3(2x + 4) as the area of a rectangle.

The rectangle has length 2x + 4 and width 3.

Use algebra tiles to label Use the labels as a guide. the length and width. Fill in the rectangle.

Six x-tiles and twelve 1-tiles have a combined area of 6x + 12. So, 3(2x + 4) = 6x + 12

We can also determine 3(2x + 4) using paper and pencil (algebraically). Each term in

the brackets is multiplied by the constant term 3 outside the brackets. This process is

called expanding.

This illustrates the distributive property.

When we use the distributive property to multiply a polynomial

by a constant term, we expand the product.

It is also possible to multiply a variable monomial and a polynomial using the distributive property using algebra tiles.

There are 2 ways to expand 3x(2x + 4).

Make a rectangle with width 3x and length 2x + 4.

Fill in the rectangle with tiles.

Connect the Ideas

We use six x2-tiles and twelve x-tiles. So, 3x(2x + 4) = 6x

2 + 12x

3x(2x + 4)

Multiply each term in the brackets by the term outside the brackets.

When we cannot use algebra tiles, we use paper and pencil.

Ex. Expand. )324(2 2 xxx

Ex. Write the product modelled by each set of tiles. Determine the product.

a) b)

Ex. Expand each of the following.

a) 2(3 4)x b) 23( 8)x

c) 2 (4 )x x d) 26 ( 3 4 2 )a a a b

e) 3 ( 2 3 )m m n f) 2( 5) 3( 2)x x

g) 2( 2 1)p p p h) 23 ( 5) (2 )m m m m

i) 4( 3) 2(2 1)x x

j) 3[2 5(2 1)]k

Pg. 166 # 1, 4 - 9

MPM1D Lesson 2.8 More Expanding Simplifying and Evaluating

Ex. Expand, simplify and evaluate if x = 2 , y = -1, z = 3.

a) 24132 yy b) 54234123 xxxxx

c) 2362 z d) 4122 x

Ex. Simplify then evaluate if a = 2 and b = -5.

a) 4 2 3

2 2

( )( )ab a b

a b b)

3

2

(3 )( 4 )

(2 )

a ab

ab

WS 2.8

Pg. 167 # 9, 15

MPM1D Lesson 2.9 Algebra – Perimeter and Area

Ex. 1 a) Write a simplified expression for the perimeter of thr triangle below. Evaluate the expression for:

b) Evaluate the expression for:

i) x = 2 ii) x = -3

Ex. 2 Write a simplified expression for the perimeter of the rectangle below.

Ex. 3 Find the missing dimension below.

Ex. 4 Find a simplified expression for the perimeter and area for each figure below.

a) b)

WS 2.9

MPM1D Lesson 2.10 Review for Unit 2 Test

Ex. 1 Expand and simplify (where possible)

a) 9x2 b) y5x2y3x c) b5a3b4a2

Ex. 2 Simplify, then evaluate (where a = 2 and b = -5):

a) ba2ba52 b) 2

3

ab2

ab4a3

WS 2.10 – Review for Unit 2 Test

Answer the following in your notebook:

1. Evaluate each of the following where 3a and 4b .

a) b3a5 b) 2b2a3 c) b5a6b3a7

2. Evaluate each of the following where 3a and 4b .

a) )ba3()b3a(2 b) )a12b5()ba4(3

3. Evaluate each of the following where 2a and 3b .

a) )a3(a4)1a3(a2 b) )a12b5(b2)b8a5(a3

4. Evaluate each of the following where 2x and 3y .

a) xy6

yx12 2

b)

75

87

yx14

yx42

c)

73

2354

yx9

yx6yx3

d)

2

43

xy4

xy4x2 e)

x5y2x3x8

y3x2y3x4

ANSWERS:

1.a) 27 b) 23 c) -5 2.a) 43 b) 32 3.a) 60 b) -150 4.a) -4 b) -36 c) 32 d) -72 e) -2

Textbook HW:

Pg. 174 # 3, 4, 7 – 20.