Date:

5.1 The Language of Mathematics MathLinks 9, pages 174-182

Key Ideas Review

Choose from the following terms to complete the statements in #1 to 3.

b inomia l exponen ts h ighest m o n o m i a l po lynomia l

symbo ls t r i nomia l var iab les

1. A lgebra uses . .., o f ten le t ters , to represent u n k n o w n numbe rs

or quant i t ies . These u n k n o w n values are called

2. A is made up of t e rms . Some of these express ions have

special names , depend ing on the number of t e rms they have.

• A

• A

• A

has one t e r m ,

has t w o te rms ,

has th ree te rms .

3. Each algebraic t e r m has a degree, which you can f ind by adding the

of the var iables in the t e r m . A po lynomia l has the

same degree as its - d e g r e e t e r m .

Check Your Understanding

4. For each express ion , ident i fy the

n u m b e r of t e rms and s ta te whe the r

it is a m o n o m i a l , b inomia l , t r i nom ia l ,

or po lynomia l .

a) 2x - 5

b) 10

c) 3z 2 - 6z + 7

5. For each express ion , s ta te the n u m b e r

of t e r m s and the expression 's degree.

a) ef f gh

b) g2~3g

c) 10

d) b2 - ab - 4d + e 2 d) 3s?t - 2

58 MHR • Chapter 5 9 7 8 - 0 0 7 - 0 9 7 3 4 4 - 2

6. Refer to the fo l lowing po lynomia ls to

answer the quest ions below.

4c 2 - 3c + 2 4ab

2f~~4 - 1 2

5p 2 - r g + h + j

Which of the above po lynomia ls

a) are t r inomia ls?

b) have a degree of 2?

c) have a degree of 0?

d) are monomia ls?

e) have a coef f ic ient of 4?

7. Wr i te the express ion represented

by each set of a lgebra t i les. Shaded

t i les are posi t ive and wh i te t i les are

negat ive .

! • • •

Date:

8. Sketch a mode l t h a t represents the

po l ynomia l .

a) x2 + 3x - 2

b) -x2 ~ 2x + 1

9. Wr i te an a lgebra ic express ion fo r each

of t he fo l l ow ing :

a) t he sum of 7 and x 2

b) t he d i f ference of 3x and 9

c) t he p roduc t of x and 4

10. Use the g iven var iab les to wr i te each

s t a t e m e n t as an algebraic express ion .

a) I f n is a number , t he product of the

n u m b e r and 5

b) I f w is t he w id th of a rec tang le and

its length is 5 cm more than its

w i d t h , t he area of rectangle

c) I f x is the n u m b e r of k i l omet res ,

the cost of ren t ing a car, in dol lars,

if the charge is $40 plus $0 .80 per

k i l omet re

5 .1 The Language of Mathemat ics • MHR 5 9

N a m e : D a t e :

BLM 5 -5

Section 5.1 Extra Practice

1. For each expression i) identify the number of terms

ii) identify the expression as a monomial , binomial, or tr inomial

a ) -2x2 i) __ ii)

b) a + b2 + s i) ___ ii)

c ) y - 5 i) ii)

d) 3d2 - Sxy i) ii)

e ) r i) ii)

f ) b2 - 2b + 7 i) ______ ii) __________________

2. Ident i fy each polynomial below as a monomial , binomial, or t r inomial . I f it is none of these, identify it as a polynomial.

c + d 3y -7e2 - 4f a2 - 3n - 6a - 5n2

x2 m2 - n - 8 a + 2b - 2c - 3d 4z 2 - y2 - 6

Monomials Binomials Trinomials Polynomials

3. For each expression i) identify the number of terms ii) state whether the expression is a monomial , binomial, or tr inomial

a ) 6t i) ii)

b) x2 + 3y - 2 i) ii)

c ) 9 - r i) ii)

d) a - 2b + 4ab i) ii)

e ) -cc/ i) ii)

f ) 5s 2 - st i) ii)

4. State the degree for each of the polynomials in # 3 . a ) _____ b) c )

d ) e ) f )

Copyright © McGraw-Hill Ryerson, 2009

N a m e : D a t e :

5. For each polynomial i) state the degree ii) state the number of terms

a ) f + g + h

b) m2-mn + n2

c ) x - y

d) s2

e ) 31

f ) 5d2 + dh - llh2 + 3

6. Write the expression represented by each set of algebra ti les.

— = positive 1-tile • = negative 1-tile

= positive x- t i le = negative x- t i le

(continued)

= positive xz - negative x'

a ) 1 • • •

7. For the polynomial 3a 2 - 4ac - 8 state the fol lowing. a ) Number of terms b) Coefficient of the first term

c ) Coefficient of the second term d ) Number of variables ________

e ) Degree of polynomial f ) Constant term ___________

Copyright © McGraw-Hill Ryerson, 2009

Verbal expression Algebraic expression

Variables Constant

Number of terms Evaluating for s = 4 and t = 2

five less than the product of two numbers s x t - 5

s and f 5 2

4 x 2 - 5 - 8 - 5 = 3

Circle each constant and underline each variable.

1. d - 6 2. 0.78 - a 3. -j" +b

Write an algebraic expression for each verbal expression.

4. the product of x and 7

5. the total of a, b, and 4

6. a number c increased by 5

7. 3 less than w

Write a verbal expression for each algebraic expression.

8. x - 2

9. y 3

10. s + r

11. cd

Write an algebraic expression for each verbal expression.

12. 3 times a number decreased by 4

13. 2 increased by twice a number

14. the total of two numbers and 7

15. the product of two numbers less 7

Evaluate for a - -2 and b = 0.3.

16. a - b 17. b - a - 2

18. 3a 19.5fc

20. ab + b-1.

21. 2fc + 3fl

Complete the table.

Algebraic Number of Value Value Expression Terms V = 3 y = - 2

22. y + 4 23. 2 y - i 24. 6 - y 25. 4y

Evaluate each expression for s = —3 and t = 2. ITiew, note */ ft* answer in parentheses is correct incorrect. Circle the letter in the appropriate column. Read first down the correct column and then down the incorrect column to decode a message.

26. 3s

27. 3f

28. s + t

29. 2f + s

30. 2s + 2t

31. st + 4

32. s + 4.5

Evaluate for x =

33. x + y + z

34. x 4- y + 0.8

35. xyz

36. x + 2z

37. ary + xz

Correct Incorrect

( -9) R A

(6) I S

(-5) L T

(-4) B O

( -2) G K

(1) C N

(1.5) H E

y = - 1.5, and z = 0.5.

Date ;

5.2 Equivalent Expressions MathLinks 9, pages 183-189

Key Ideas Review

1. Comple te the fo l lowing s ta temen ts .

a) In the monomia l Sab, t he var iables are and

b) In the m o n o m i a l - 7 w x 2 , the coef f ic ient is . The var iables are w and x.

The exponen t for w is and the exponen t of x is

c) For the monomia l 18, is there a coef f ic ient or var iable? YES NO

2. In the th ree like t e r m s below, circle w h a t is alike a m o n g t h e m . Then , comb ine

the t e r m s .

3x 2 ~Ax2 - x 2 Combined t e r m :

3. Are the t e r m s below like te rms? YES NO Expla in.

5x 5x 2 5y

Check Your Understanding

4. For each of the fo l l ow ing , s ta te the

value of t he coef f ic ient . T h e n , s ta te

the n u m b e r of var iables for each t e r m .

a) y b) -3b2

c) 6sr d) - 1 5

e) -dh f) be

5. Use the fo l lowing monomia l

expressions to answer the quest ions

below.

-cd 9 r 4 x k2 -xy -3jk

a) Which have a coef f ic ient of - 1 ?

b) Which have two var iables?

c) Which have a coef f ic ient of 1?

d) Which have only one var iab le , w i th

an exponen t of 1?

6 0 MHR • Chapter 5 9 7 8 - 0 0 7 - 0 9 7 3 4 4 - 2

Date:

6. Circle the l ike t e r m s in each g roup .

a) 14 3r -r2 -r 3s

b) -Ay 8xy 2x 0 . 3 / Y 2

0 12c cd 1.2d 6cd cd2

7. Rearrange the po lynomia l by g roup ing

l ike t e r m s .

a) 9 - 5c - 8 + 5c 2 + c - c2

b) 8 m - 9 + 2 m 2 + 6 + 3 m 2 - 6m

c) -3d2 + 3 ^ - 2 + 6 ^ - 8 ^ + 7

10. a) Draw a f igure w i t h a pe r ime te r t ha t

is represented by

(s) + (2s) + (s + 5) + (3s ) ,

where each va lue in parentheses

represents t he length of one s ide.

Label each side l eng th . Explain why

you made each side t he length tha t

you d id .

b) S impl i fy the express ion for t he

pe r ime te r by comb in ing l ike t e r m s .

8. Rearrange each po lynomia l by

g roup ing l ike t e r m s . Then , s impl i fy by

add ing or sub t rac t ing .

a) -b2 + 6 + 5b2 - 8 + 9

b) It + 14 + 6r - 5 - 3r 2 + 4 t 2

11. A mechanic charges $70 an hour plus

the cost of par ts to repai r a vehic le.

The par ts cost $215 fo r the repai r on

Tamara's car.

a) Wr i te an express ion fo r the to ta l

cost , C, of repa i r ing Tamara's car

fo r any n u m b e r of hours , n.

c) 5n - 3n2 - 7 + 9n + 3 - 2n2

d) 3y2 + 4 - 6y2 - 6 + 3y - 5 + 2y

9. Wr i te and s impl i fy an express ion

fo r t he pe r ime te r of the t r iang le by

comb in ing l ike t e r m s .

6 + 3

b) Use the express ion you created

in par t a ) to calculate the cost of

repairs t h a t take 3 ^ h.

5.2 Equivalent Expressions • MHR 61

N a m e : D a t e :

Section 5.2 Extra Practice BLM 5-7

1. Determine

i) the value of the coefficient ii) the number of variables for each term a ) -t i) ii) ________ b) 4d2 i) ii)

c ) 12 i) ii) d ) -8de i) ii)

e ) b i) ii) f ) - c 2 i) ii)

2. Match the expression with its description by placing the correct letter in the blank.

A -4x a constant

B 17 a binomial with two variables

C lab - 1 is the coefficient

D 3y2 - 2y - 4 is the coefficient

E -m a binomial with a degree of 2

F 5x - 3y a monomial with a degree of 2

3. Circle the like terms in each group.

a ) 4x, 4y, x2, -x, y2 b) 6, 2x, -2.5, 3y, - 0 . 1

c ) a, 4b, -3ab, la, 1.5a d ) -f, 3ef, f 2 , -6fz, 5e

e ) 6st, -10s, —st, -st, t f ) pq, - 0 . 6 p 2 , 5q, - p 2 , 10p 2

g) 0.5jk, -jk, j2, 6jk, -k h ) | , | r , 0.12, r2, 9

4. Collect like terms.

a ) 3m - m2 - 6 4- 3 m 2 b) -4k - k2 + 5k - 7k2 + 8

t ) -c - c2 + 3c + c2 d ) 7 - 10 + 5n - n + 9 + 8n

e ) -2b2 - 7b + 36 2 - 8b + b f ) w2 - 3w - 8w2 + 7w2 + lOw

g) -2a - 1 - a - 7 - 5a h ) 3s + 6 - 6s 2 - 8 + 7s - 2s:

Copyright © McGraw-Hill Ryerson, 2009

N a m e : D a t e :

BLM 5-7 (continued)

5. A rectangle's length is 7 cm greater than its width, w. a ) Draw the rectangle and label its dimensions.

b) Write the expression to find its perimeter.

c ) Collect like terms.

6. The cost of publishing the school yearbook was $440. The yearbook commit tee priced the yearbook at $8. a ) Write an expression that represents the profit, p, for the number of

yearbooks sold, n.

b) How many yearbooks need to be sold for the yearbook commit tee to break even?

Copyright © McGraw-Hill Ryerson, 2009

5.2c-d Applications

Like terms have the same variable raised to the same exponent, r, 4r, lOlr Unlike tenns have different variables or the same variable but different exponents. 7b, -3a, x2, x

Simplify.

1. 3y + 4y

3. -4b - b

5. 2 s 5 + 3s s

Simplify.

7. 2a - 3a + 5a

9. 0.4r + 0.5r + O.lr

10. r2 + 2^ + 3T2

Simplify.

11. -3a2 + 2b2 + 3a2

12. -4e + 2d + 3e

13. 5s 3 + 2s3 - s 2

14. - 4 x - 2y - 2a

2. 7a2-2a2

4. 5p + (-2p)

6. c2 + c2

8. - 3 c - 2c - c

Simplify.

20. 2c + 3 + 4 d - c + d

21. x - 2 + 2y + 5 - y

22. - 2 + 2z + (~3w) + 4 + 2

23. 3 + 4x2 + y 2 + x 2 - l

Write an expression for each perimeter in 2 different ways.

29.

+ 0.5s

1.5s

30. 2r

3r

31.

Date:

5.3 Adding and Subtracting Polynomials MathL 'mks 9, pages 190-199

Key Ideas Review

1. Which equat ion does the algebra t i le mode l represent?

A ( 4 x - 4 ) + (x + 3) = 5x - 1

B ( 4 x + 4 ) - ( - x + 3) = 5x + 1

C ( 2 x - 2) + ( 3 x + 1) = 5 x - 1

• 2. One word can replace the quest ion marks in the fo l lowing sentences: The ? of

a po lynomia l is found by tak ing the ? of each of t he t e r m s . To sub t rac t

po lynomia ls , you can add the ? .

The word is

Check Your Understanding

3. Add the po lynomia ls .

a) ( 6y - 4 ) + ( 2 / + 2)

b) {b2 + 5) + \-2b2 - 3)

c) ( - 3 s 2 + 7s) + ( - s 2 - 6 )

D (2x - 2) - ( - 3 x - 3) = 5x + 1

5. Which of t he s ta temen ts do the

algebra t i les represent?

• • • A ( x 2 + x - 3 ) + ( x 2 - 2x + 3)

4. Perform the ind icated opera t ion . Then , B ( X 2 + x _ 3 ^ + (_X2 _ 2x + 3) s impl i fy by combin ing l ike t e r m s .

a ) ( 8 + 5d) + ( - d - 9 ) c ( x 2 _ x _ 3 ) + ( _ x 2 _ 2 x + 3 )

W (_4m; - 4) + (_2m» - 1) D (x2 + x + 3) + - 2x + 3)

c) ( - 6 r 2 + 3 r - 7) + (Sr 2 - 2 r - 2)

6 2 MHR • Chapter 5 9 7 8 - 0 0 7 - 0 9 7 3 4 4 - 2

Date:

6. Give the opposi te of the express ion.

Express you r answer using both

d iagrams and symbo ls .

a)

b)

9. Consider the t r iang le below.

2x + 2

x + 3

a) Wr i te the uns impl i f ied express ion

for the per imeter .

b) S impl i fy the express ion f r o m par t a)

by comb in ing l ike t e r m s .

7. Wha t is the opposi te of each

expression?

a) -3y2

c) I f t he pe r ime te r of t he t r iang le is

25 c m , calculate the value of x.

Veri fy t h a t you r answer is correct .

b) 6g - 3

c) 2b2 - 4b + 7

d) -4cf -3d - 6

e) -k2 - 8k + \

8. Change the subt rac t ion operat ion to

add ing the oppos i te . T h e n , combine

l ike t e r m s .

a) ( 3 r - 5) - ( 5 r + 2)

b) (6 - 3 0 - (4 - 5f)

c) (-4n2 + 5) - (~n2 - 9 )

Jose, Tyler, and Mike spli t some money

they made work ing on the weekend.

They each worked a di f ferent number of

hours, so they have to split the money

fairly. Jose receives twice the amount

tha t Tyler receives, and Mike receives

$10 less than Tyler. Let x represent the

amoun t tha t Tyler receives.

a) Wr i te the express ion tha t

represents the to ta l a m o u n t t ha t

t hey receive.

b) S impl i fy the express ion in par t a)

by comb in ing l ike t e r m s .

d) ( 6 a 2 + 2a - 5) - ( 4 a 2 + 5a + 7)

5.3 Adding and Subt rac t ing Polynomials • MHR 6 3

N a m e : D a t e :

Section 5.3 Extra Practice BLM 5-9

1. Add the polynomials by collecting like terms. Then, simplify.

a ) (3x 2 -2x) + (x2 + x) b ) ( 4 n 2 -2n - 4) + (-n2 + 5n) c ) (7r - 8) + ( 3 A 2 - 11) d ) (2b2 - Sb) + (-2b2 + lib) e ) (7L 2 - 6 t + 9) + ( - 2 1 2 + 6r - 5) f ) (-14k - 10) + (Sk -23)

2. Determine the opposite of the expression represented by each diagram. Express the answer in diagrams and symbols.

• = positive 1-tile • = negative 1-tile

= positive x- t i le i i = negative x- t i le

= positive x2 = negative x2

a )

• b )

• • • 3. Determine the opposite of each expression,

a ) 6a b ) -3c2 - 9 c ) d2 - 8d + 2 d ) 6w2 + 4w - 0.8

4. Subtract the polynomials by adding the opposite terms, collecting like terms, and then simpl i fy ing. a ) (5a - 4) - (3a - 2) b ) (7 - 6r) - (3 + r ) c ) (6y2 - 2y) - (-y2 - 3y) d ) (8 - St) - (-9 - 4t) e ) (h - 1) - (3h2 + 7) f ) (4k2 - 6k + 1) - (-2k2 + 5)

5. A tr iangle has the dimensions shown.

3x - 9

a ) Write the unsimplif ied expression for the perimeter of the tr iangle. b ) I f x = 6, what is the perimeter? Show your work. c ) Simplify the expression in part a) for the perimeter of the tr iangle. Show

your work. d ) Use the simplified expression to verify the perimeter when x = 6. Show

your work.

Copyright © McGraw-Hill Ryerson, 2009

5 3 a Add ing Polynomia ls

1 • Use algebra tiles to model each sum of

binomials. Record your answer symbolically.

a) (5g + 3) + (2g + 4)

b) (3 - 2j) + ( - 4 + 2j)

c) ( p + 1) + (5p - 6)

d) (7 + 4m) + (Sm + 4)

2. Use a personal strategy to add.

a) (6x + 3) + (3x + 4)

b) (5/? - 4) + (2b + 9)

c) (6 - 3 / ) 4 ( - 3 - 2y)

d) ( - « + 7) + (3« - 2)

e) ( --45 - 5) + (6 - 3s)

f) (1 - 7/7.) + ( -7 /z - 1)

g) (8m + 4) + ( - 9 + 3m)

h) ( - 8 m - 4) + (9 - 3m)

3. Add. Which strategy did you use each time?

a) (4m 2 + 4m - 5) + (2m 2 - 2m + 1)

b) (3fc2 - 3 k + 2) ! (-3k2 - 3 k + 2)

c) (-7p ~~ 3) + (p2 + 5)

d) (9 - 3f) + (9t + 3? - 6l)

e) (3x 2 - 2x + 3) + (2x 2 + 4)

f) (3x 2 - Ix + 5) I (6x - 6x 2 + 8)

g) (6 - 7x + x 2 ) + (6x - 6x 2 + 10)

h) (1 - 3r + r 2 ) -f (4r + 5 3 ^ )

4. A student added (4x 2 - 7x + 3) and

(—x 2 — 5x + 9) as follows.

(4 * : 2 _ 7 ? ^ ^ ( i ^ - ^ . ^1 )

^ ^ - Z ^ . L —

5. a) For each shape below, write the

perimeter:

• as a sum of polynomials

• in simplest form

n + 5

2/7+5

ii)

iii) 6f+5

f 2r+1

iv)

f+2

3/+1

5 3 c Subtract Po lynomia ls

1. Use a personal strategy to subtract.

Check your answers by adding.

a) (3x + 7) - ( - 2 x - 2)

b) (b2 + 4b) - (-3b1 + 7b)

c) { 3x + 5) - (4x4- 3) d) (4 - 5p) - ( - 7 p + 3)

e) (6x2 + 7x + 9) - (4x 2 + 3x + 1)

f) (12m 2 - 4m + 7) - (8m 2 + 3m - 3)

g) ( 4 x 2 - 3x - 11) - (x 2 4x - 15)

h) (1 - 3r + r 2 ) (4r 4 5 - 3r 2)

2. A student subtracted like this:

^ yz - z y ^

a) Explain why the solution is incorrect.

b) What is the correct answer?

Show your work.

c) How could you check that your answer

is correct?

d) What could the student do to avoid

making the same mistakes in the future?

3. The perimeter of each polygon is given.

Determine each unknown length,

a) 6w 4- 14

2w+ 3

2w+ 3

b) 75 4- 7

3s + 2

c) lOp 4- 8

3s+ 2

p + 3 p + 3

4. The diagram shows one rectangle inside

another rectangle. What is the difference in

the perimeters of the rectangles?

2x+1

4x+3

Vocabulary Link

Unscramble the letters of each term in column B. Use the clues in column A to help you. Each

term is one to four words long.

A B 1. an algebraic expression made up of terms connected

by the operations of addit ion or subtract ion;

for example, 3x- - 4

LYPNAOOIML

2. terms that differ only by their numerical coefficients,

such as3x and -•• 2x . .. .

STEMILKER

3. an expression formed f rom the product of numbers

and/or variables, such as 9x

MRTE

4. a polynomial w i th three terms LIOITRMNA

5. a polynomial with two terms OMLNIAIB

6. in algebra, te rms are often arranged in this order GDCSDENINE

7. a polynomial wi th one te rm IALNOOMM

8. branch of mathemat ics that uses symbols to represent

unknown numbers or operations

EABLGRA

9. the sum of the exponents on the variables in a single t e r m ; for example, for 3xz, it is 2

EMARDGETEORFE

10. the degree of the highest degree te rm in a polynomial ; for example, for 8bJ - lb, it is 2

LYPNAOOIMLEEEFOAGRD

Chapter 5: Vocabulary Link • MHR 6 5