Investigating Monomials (pp. 1 of 4)

Investigating Monomials (pp. 1 of 4) 

Algebra 2 

HS Mathematics 

Unit: 06 Lesson: 01 

Definition 

1) Monomial—What is a monomial? 

The following are monomials: 

5 

2 3 – 8.5 

So monomials can include: 

2x 

5 2 

y 

3 

8.5xy 

2 

2 3 7 

0 

9a b c 5 

However, these are not monomials: 

So a monomial cannot have: 

2 + x x 

2 8. 5 

9 

x 

x 

Vocabulary 

Using the monomials above, identify examples of the following definitions. 

2) A ___________________ is a monomial with no variables, 

Examples: 

3) In a monomial, the ___________________ is the numeric factor of the variable (or variables) 

Examples: 

4) The _____________________ of a monomial is the sum of the exponents on the variables only 

Monomial 

Monomial 

Examples: 

2 3 7 

2x 1 9a 

b c 

3 

2 

5 

y 5 

2 

8.5xy 

0 

©2010, TESCCC 08/01/10 page 11 of 61


Algebra 2 



Simplifying Monomials 

Multiplying 

Sample: 

2 3 4 

(5x 

y )(4xy 

) 

Expand: 5 x x y y y 4 x y y y y 

Re-order: 5 4 x x x y y y y y y y 

Simplify: 

Dividing 

Sample: 

4 3 

18x 

y 

2 

6xy 

Expand: 

18 x x x x y y y 

6 x y y 

Simplify: 

 

What’s the short-cut? 

the coefficients 

 

What’s the short-cut? 

the coefficients 

the exponents 

the exponents 

 

What’s the rule? 

a 

m a n 

 

Note: Bases must be the same. 

 

What’s the rule? 

m 

a 

 

n 

a 

Note: Bases must be the same. 

Other Rules 

Samples 

a 

n 

1 

 

n 

a 

2 

5 

 

b = 

b 

3 

4 

x = 

9 

a m 

n a mn 

( 5 

2 ) 3 

 

(x 4 ) 5 = 

( 

 

b 2 ) 6 = 

a 

 

b 

m 

a 

 

b 

m 

m 

5 2 

 

8 

7 

x 

 

y 

= 

b 

 

c 

3 

2 

 

 

 

5 

= 

( 

m m 

ab) 

a 

b 

m 

2 2 

(5 x ) 2 ) 6 

2 4 3 

(xy 

( 4x 

y ) 



Algebra 2 



Sample Problems 

1) 2 3 3 

2ab 

4a b c 

2) 2 

3x 

3 

3) 

11 

x 

x 

5 

4) 2 3 

6x y xyz 

3 

5) 3 2 

3 

 

6a b 

6) 

2 

3 6 

3x y 

5 2 1 

xy 

z 

7) 

2 6 

A rectangle has a width represented by 3x y and a length represented by 

expression can be used to represent the area of the rectangle? 

5 3 

8x 

y 

. What 

5 3 

8x 

y 

2 

3x 

y 

6 

8) 

4 

The area of the triangle below is represented by 14x 

y 

the base of the triangle. 

9 

. Find the expression that represents 

2 5 

7x 

y 

b = ? 



Algebra 2 



Practice Problems 

Simplify the following expressions. 

3 3 5 

1) x x x x 

2) 2 

3 

3xy 

3) 

6 6 

x y 

4) 

2 6 2 6 

3 x y 

4x 

y 

x 

3 

y 

4 

5) 

4 6 

ab c 

6) 

a 

5 

bc 

2 

5 3 

2a 

b c 

3 3 

8a 

b c 

3 

7) 

40a 

20a 

1 

7 

b 

8) 5 8 3 2 2 

 

15m 

n m 

n 

5 

3 

4 

b 

45m 

n 

9) The height of a triangle is represented by the expression 

2 3 5 

8p 

q r 

6 

15p 

qr 

3 

. The base is represented by 

. Find the expression that can be used to represent the area of the triangle. 

10) The length and area of a rectangle are given in the diagram below. Find the expression that can 

be used to represent the width of the rectangle. 

w = ? 

Area = 72m 15 n 10 

4m 3 n 7 


Investigating Polynomials (pp. 1 of 3) 

Algebra 2 



Vocabulary 

Polynomial – ____________________________________________________ 

A polynomial can be classified according to how many “terms” it has. 

Category Sample Definition 

x One term 

5 2 3 

y 

2 

4x 2 9x 

_________ terms 

2x 2 3x 

1 

_________ terms 

4 

2 

3a b 7bc 

6cd 

8 _________or _________ terms 

Degree of a Polynomial – _________________________________________ 

3a 

4 

Sample Degree of each term Degree of Polynomial 

x 5 5 

5 2 3 

y 

2 

4x 2 9x 

2x 

2 3x 

1 

b 7bc 

2 

6cd 

8 

Operations To simplify polynomials with addition, subtraction, and multiplication: 

Clear grouping symbols using properties of algebra (distributive) 

Combine like terms using properties of algebra (commutative, associative). 


1) 

2 

2 

2 

2 

( 4x 

3xy 

5y 

) 2(3x 

3xy 

5y 

) 2) 

2 

2 

2 

2 

(3x 

6xy 

7y 

) ( x 

5xy 

2y 

) 



Algebra 2 



Perform the indicated polynomial operations. Simplify answers, and classify each answer by its 

degree and number of terms. 

3) 

2 2 

3 

3xy (5x 

y 6xy 

) 

4) ( x 8)( x 3) 

5) 

2 

( x 4) 

6) ( 2x 

3)(2x 

3) 

Applications: 

7) A rectangle has a width represented by 4x + 5 and a length represented by 3x + 2. 

What expression can be used to represent the area of the rectangle? 

8) The diagram below shows an isosceles triangle. Find the expression that represents the 

perimeter of the triangle. 

3a + 5 

3a 2 + 2a –1 



Algebra 2 




Simplify the following polynomials. 

2 

2 

2 

2 

1. 2(4x 

3xy 

7y 

) (2x 

5xy 

3y 

) 

2 

2 

2 

2 

2. (5m 

2mp 

6p 

) 2( 3m 

5mp 

p ) 

3. 2x( 

x 5) x(3 

x) 

2 2 

4. 5a 

b(7ab 

3a 

4b) 

5. ( 3x 

2y)( 

4x 

y) 

6. 

( 2x 

1) 

2 

7. ( 5x 

4)(5x 

4) 

2 

8. (2n 

3)( n 5n 

1) 

9. The height of a triangle is represented by the expression (x + 2). The base is represented by 

(2x – 8). Find the expression that can be used to represent the area of the triangle. 

10. The width and length of a rectangle are given in the diagram below. Find the expression that 

can be used to represent the area of the rectangle. 

2 3 

5x y 

4x y 


Algebra 2 



Operating Around Polynomials 

Work problem exercises A and B, then connect the answers of A1 to B1, A2 to B2, etc. with a line. Continue until all 

problems are completed. Shade in areas with odd numbers. 

Exercise A - Simplify 

1. (-3a + 2b – c) + (4b + 3a – 2c) 15. (2x 3 – 3x 2 y + 2xy 2 ) – (-4x 2 y – x 3 ) 

2. (2x 2 y 2 – 4a 2 y + a 2 b 2 ) + (4a 2 y – 3a 2 b 2 ) 16. (6xy 2 – 3x 2 + xy) – (xy – 3x 2 ) 

3. (7b – 4a + 2c) + (-7a + 4b + 3c) 17. (xy + x 2 y) – (3xy – 5x 2 y) 

4. (6x 2 – 2x + 3xy) + (2x – 6x 2 – xy) 18. (xy + ab)(2xy – 2ab) 

5. (6x 3 y 2 – 2xy – x 3 y) + (x 3 y + 4xy – 6x 3 y 2 ) 19. (a 2 b 2 – 3)(ab 2 + 2) – a 3 b 4 

6. (5x 2 + y 2 – xy) + (x 2 – 2y 2 + 2xy) 20. (3m 5 – ac)(3m 5 + ac) 

7. (2y 3 – 3xy) + (x 3 – 3y 3 + 3xy) 

8. (7x 3 + 2xy 2 ) + (-4x 3 + x 2 y) 

9. (9a – 3b – 2c) – (9a – 3b – c) 

10. (4x 2 – 2y 2 + xy) + (2x 2 + y 2 ) 

11. (4a – 2b + c) – (6b + 4a + c) 

12. (4a 2 c 2 + 6m 10 ) – (-3m 10 + 5a 2 c 2 ) 

13. (4a – 2c + 3b) – (4a + 3b) 

14. (9a – 3c + 4b) – (11a – 3c + 4b) 

Exercise B - Simplify 

1. (2a – 4c + 5b) + (2c – 5b – 2a) 

2. (6b – 2a + c) + (a – b – c) 

3. (a 2 b 2 – 4ab 2 – 6) + (a 2 b 2 + ab 2 ) 

4. (5x 2 – 3x + 2) + (3x – 2 + 5x 2 ) 

5. (3a 2 – 9ab) + (9b 2 – a 2 ) 

6. (9a – 3b + 5c) + (-20a + 14b) 

7. (5a – 3c) + (c – 5a) 

8. (9ab – 3b) + (-3c + 9b – 9ab) 

9. (a 3 b – a 2 b + a 3 ) + (a 2 b – b 3 – a 3 b) 15. 3c – (2a + 3c) 

10. (9b 2 – 3ab + a 2 ) – (6ab – a 2 ) 16. 10x 2 – y 2 – (-y 2 ) 

11. (3xy + x 2 y) – (5xy – 5x 2 y) 17. (-3ab 2 + a 2 b 2 ) – (6 – a 2 b 2 ) 

12. 3b – (a – 2b) 18. (a – 3b)(2a – 3b) 

13. 6b – (2a + 6b) 19. (x + y)(3x – 3y) 

14. (4b – 2c) – (12b – 2c) 20. (x – y)(x 2 + xy + y 2 ) 


“Fact”ors About Islands 

Algebra 2 



Multiply factors to determine the original polynomial. Use your solutions to fill in the blanks below. 

Code 

Original Polynomial 

Letter 

Factors 

2 

1. 49 

4 x 6 x 2 

x A 

2 

2. 3x 

17x 

10 

3. 

4. 

5. 

6. 

7. 

B 2 

2 

7 63 

y 

x 3y 

x C x 2 1x 4 x 

2 1 

2 

8y 

16 

D x 7x 

7 

2 

6 -60 150 

x x E 2 

4x 

81y 

2x 7 

2 

 

2 

H 4ab a 3ba 3b 

2 

-6 9 

2 

I 3x 75x 

4 

x x x K 4x 4yx 

4y 

2 

x x L 2x 9y2x 9y 

x M 3x 22x 

3 

2 

x x N 7x 

3x 

3 

2 

y y O 3x 2x 

5 

x xy y 

8. 

3 2 

9. 

10. 

11. 

12. 

13. 

14. 

15. 

16. 

17. 

18. 

19. 

4 5 12 15 

6 5 6 

3 

2 16 

4 32 48 

14 13 12 

x 

y 

P 2x 2x 2 2x 

4 

10 10 

6 

x 1 

R 7y 

42y 

3 

4x 

64y 

S 5x 3y25x 2 15xy 9y 

2 

 

2 2 

2 

15x 

23x 

28 

T 2 

125x 

27y 

6 x 5 

3 3 

2 

U 4x 

5x 

3 

x V x 5 y 5 x 5 y 

5 

 

2 

4x 

28 49 

4ab 

36ab 

Y 2 

3 3 

y 4 

No man is an island – but we should know all their names. 

___ ___ ___ ___ ___ ___ ___ ___ (Canada) 

___ ___ ___ ___ ___ ___ ___ ___ (Japan) 

13 16 14 5 2 12 16 11 19 2 15 15 11 16 1 2 

___ ___ ___ ___ ___ ___ ___ ___ (Portugal) 

___ ___ ___ ___ ___ ___ ___ (Russia) 

9 11 1 18 16 12 11 17 15 8 12 16 6 18 17 

___ ___ ___ ___ ___ (Philippines) 

___ ___ ___ ___ (United States) 

10 11 3 11 4 9 11 8 16 

___ ___ ___ ___ ___ ___ ___ ___ (United Kingdom) 

19 18 7 12 16 1 18 17 


Factoring (pp. 1 of 4) 

Algebra 2 



Review 

Try these items from middle school math. 

A) What numbers are the 

factors of 24? 

B) Write down the prime 

factorization of 72. 

C) 

36 

Simplify 48 

using the 

greatest common factor 

(CGF). 

 

What does it mean to “factor” a number (or, to “find its factors”)? 

Polynomials: 

2 

2 2 

2 

( 2x 

3y)(4x 

y) 

8x 

2xy 

12xy 

3y 

8x 

14xy 

3y 

 

Consider the polynomial problem above. What are the “factors”? 

Look back at the previous activity (“Fact”-ors About Islands). What are the factors of x 2 49 ? 

 

What does it mean to “factor” a polynomial (or, to “find its factors”)? 

Factoring Polynomials 

Follow these steps to factor polynomials. 

Step One: Always look to see if the terms have a greatest common factor (GCF) (other than 1) 

Example: 

6x 2 2 

2 

14x 

Example: 4a b 6ab 

10ab 

GCF = GCF = 

Factors: _____ ( _____ + _____ ) Factors: _____ ( _____ + _____ + _____ ) 

Step Two: 

After checking for the GCF, remaining polynomials can be factored by several different 

methods, according to the number of terms in the polynomial. 

A) Two Terms 

1. 

2 2 

Difference of Squares: a b ( a b)( 

a b) 

Example: 

9 64y 

2 2 

x Example: 20x 

2 45 

GCF = GCF = 

Factors: 

Factors: 



Algebra 2 



Two Terms (continued) 

2. 

3 3 

2 

2 

Difference of Cubes: a b ( a b)( 

a ab b ) 

Example: 

GCF = 

Factors: 

3 

x 

y 

3 

3. 

3 3 

2 

2 

Sum of Cubes: a b ( a b)( 

a ab b ) 

3 

Example: 16m 2p 

GCF = 

Factors: 

3 

B) Three Terms 

1. Leading Coefficient of 1 

2 

Example: x 4x 12 

Example: 2x 

2 10x 

12 

GCF = GCF = 

Factors: 

Factors: 

2. Leading Coefficient other than 1 

Example: 6x 

2 7x 

5 

Various Methods 

 

Guess and check 

GCF = 

Factors: 

 

Box method 



Algebra 2 



 

Gross Product 

 

Bottoms up 



Algebra 2 



C) Four Terms 

Grouping 

Example: 12xy + 2x + 30y + 5 Example: 6ab + 2a + 15b + 5 

___ ( ___ + ___ ) + ___ ( ___ + ___ ) 

( ___ + ___ )( ___ + ___ ) 


Factor the following polynomials. 

1) 

2 

2 

14x y 4xy 

2xy 

2) 4x 

2 9 

3) 

3 2 

2a 2ab 

4) x 

3 27 

5) 64x 3 1 

6) x 3 y + 8y 

7) 

2 

x 14x 49 

8) 2x 

2 6x 

56 

9) 

2 

y 3y 

54 

10) 4y 

2 11y 

3 

11) 5a 2 22a 

8 

12) 36x 

2 6x 

20 

13) 6x 2 9x 

81 

14) ab 9a 

9b 

81 

15) 4xy 

8x 

7y 

14 

16) The area of a right triangle is represented by the expression 6x 2 + 5x – 4. If the height of the 

triangle is represented by the expression 3x + 4, find an expression to represent the base. 


Tying Up Polynomials (pp. 1 of 3) 

Algebra 2 



Simplify the following polynomials. Box answers. 

1) 3 2 4 

4ac 

 

3ac 

 

3 2 

2) 

4 2 

25x 

y z 

2 2 

5x 

y z 

2 

3) 3 5 

4m 

np 

 

2 2 2 

16 

 

m n p 

4) ( 2x 

5)(2x 

5) 

5) 2 

3 4 

x 6) ( 2x 

3)( x 5) 

7) 

2 

2 

2 

2 

2(5x 

3xy 

2y 

) (3x 

2xy 

y ) 

8) 

2 

2 

(2x 

y)(4x 

2xy 

y ) 



Algebra 2 



Factor the following polynomials. Box answers. Check answers by multiplication. 

9) 9x 2 6x 

1 

10) 75x 

2 3 

11) 4x 3 32 

12) ax bx ay by 

13) 3x 2 14x 

5 

14) 3x 

2 24x 

48 

Use polynomial operations to solve the following problem situations. Box answers. 

15) An isosceles triangle has sides represented by the expression given in the figure below. What 

is the perimeter of the triangle? 

3x 2 + 5x – 4 

2x 2 –8x + 3 



Algebra 2 



16) What expression can be used to represent the area of the rectangle below? 

2x 

1 

3x 

4 

2 

17) The area of the triangle below is represented by the expression x 2x 15 

. What expression 

can be used to represent the height of the triangle? 

h = ? 

x + 5 

18) The area of a rectangle is represented by the expression 2x 

2 5x 

2 . The length of the 

rectangle is represented by x + 2. Find the width. 


Drop That Ball! 

Algebra 2 



1. If you were to collect data from a motion detector held above a 

bouncing ball, imagine the graph of the height of a bouncing ball 

versus time. Sketch the graph of your conjecture. 

2. Use a graphing calculator, data collection device, and large ball to collect data on a bouncing 

ball for approximately 8 seconds. Sketch the actual graph of the height of the ball against time 

below. 

3. Choose one of the parabolas and use the trace feature of the calculator to find the coordinates 

of the maximum point. 

(____, ____) 

4. Use your knowledge of parabolas and transformations of functions to find the equation of the 

parabola chosen in question #2. Enter the function into the graphing calculator to graph the 

equation over the scatterplot to check your work. 


Investigating Transformations on Quadratic Functions (pp. 1 of 3) 

Algebra 2 



Complete the table to make an 

accurate graph of the quadratic 

2 

parent function, y x . 

x 

-4 

-3 

-2 

-1 

0 

1 

2 

3 

4 

y 

2 

In the items that follow, use a graphing calculator to compare the graph of y x to various 

transformations of this parent function. Sketch the graphs, and discuss your conclusions. 

1. Functions Graph (Sketch) What can you conclude? 

Y 1 = 

Y 2 = 

2 

x 

2 

x 


Y 1 = 

2 

x 

Y 2 = x 

2 4 


Y 1 = 

2 

x 

Y 2 = x 

2 4 


Algebra 2 





Y 1 = 

Y 2 = 

2 

x 

2 

( x 4) 


Y 1 = 

Y 2 = 

2 

x 

2 

( x 4) 


Y 1 = 

Y 2 = 

2 

x 

2 

2x 


Y 1 = 

2 

x 

Y 2 = 0.5x 

2 



Summarize observations in the table below. 

Algebra 2 



Function Effect Domain/Range 

2 

y x 

Parent Function 

D: All real numbers 

R: y 0 

Tell how the functions below are 

different from the parent function. 

2 

y x 

y 

x 2 4 

y 

x 2 4 

y 

2 

( x 4) 

2 

y ( x 4) 

2 

y 2x 

y 

1 2 

2 

x 


Characteristics of Quadratic Functions (pp. 1 of 5) 

Algebra 2 



Here are the two forms in which quadratic functions can be written: 

Vertex Form: 

2 

y a( 

x h) 

k 

General (Standard) Form: 

2 

y ax bx c 

In these equations, a, b, and c, h, and k represent constants, but a cannot equal zero. 

Why must we say a 0 ? 

Comparison of Characteristics 

Characteristic 

vertex 

axis of 

symmetry 

Standard Form 

2 

y ax bx c 

( b 

2 a 

, ?) 

b 

Plug x = into the 

2a 

equation to find y 

b 

x 

2a 

y-intercept (0, c) 

point symmetric 

to y-intercept 

x-intercept(s) 

Vertex Form 

2 

y a( 

x h) 

k 

(h, k) 

x = h 

(0, ?) 

Plug x = 0 into the 

equation to find y 

The y-intercept (and other points) can be reflected 

across the axis of symmetry to find other points on the 

graph of the function. 

These points can be read from the graph or table. 

When in doubt, use the calculator’s CALC 2: zero 

command (2 nd , TRACE). 

The two forms of 

quadratic equations 

provide information 

about the function’s 

graph in different ways. 

However, some things 

are the same, 

regardless of which 

form you use. 

Find each characteristic for the functions described. 

2 

Characteristic y x 2x 

3 

y ( x 1) 

2 4 

vertex 


symmetry 

y-intercept 

symmetric 

point to 

y-intercept 


Compare with the results from the handout: Investigating Characteristics of Quadratic 

Functions. 



Algebra 2 




Find the characteristic parts of each function. Use this information to produce the graph. 

2 

A) y x 6x 

2 


Vertex 

Value 

x 

y 

Axis of 

Symmetry 

y-intercept 

Point 

symmetric to 

y-intercept 


B) f ( x) 

2( x 1) 

2 3 


Vertex 

Value 

x 

y 


Symmetry 

y-intercept 

Point 

symmetric to 

y-intercept 




Algebra 2 




For problems #1-4 make a table of values, graph the function, find the vertex, determine if the vertex 

is a maximum or minimum, write the equation of the line for the axis of symmetry, find the y-intercept 

and symmetric point, and give the x-intercepts. 

1) 

2 

f ( x) 

x 4x 

5 


Vertex 

Value 

x 

y 


Symmetry 

y-intercept 

Point 

symmetric to 

y-intercept 


2) 

2 

y ( x 2) 


Vertex 

Value 

x 

y 


Symmetry 

y-intercept 

Point 

symmetric to 

y-intercept 




Algebra 2 



3) 

2 

y x 

4x 

12 


Vertex 

Value 

x 

y 


Symmetry 

y-intercept 

Point 

symmetric to 

y-intercept 


4) y 2( x 1) 

2 1 


Vertex 

Value 

x 

y 


Symmetry 

y-intercept 

Point 

symmetric to 

y-intercept 




Algebra 2 



5) True Value Fabricators produces 

circular iron cast disks to be used as 

endplates for pipes. The cost of the disks 

is a quadratic function of the diameter. 

The cost of some disks is given at right. 

1 inch diameter …. $12.00 





A) In this situation, what are the independent and 

dependent variables? 

B) Sketch a scatterplot of the data. Label the axes. 

C) Enter the data into the graphing calculator. Use 

2 

transformations of y x to determine a 

representative function for the data set in 

2 

y a( x h) 

k form. 

D) What would be a reasonable domain and range for this 

function? 

E) Find each of the characteristics and explain their meaning in the problem situation. 

Characteristic Value(s) Meaning in Problem Situation 

vertex 


symmetry 

y-intercept 


F) What would be the cost of a disk with a diameter of 12 inches? 

G) If the cost of the disk is $522, what would be the diameter of the disk?

Investigating Monomials (pp. 1 of 4)

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