Lesson 5.1: Modelling Polynomials (P.210)
(updated as of Oct. 2010)
Focus: Model, write, and classify polynomials
Complete Investigate (p.210)
Algebra Tiles (only positives)
Algebra Tiles (ans incl)
Algebra Tiles (describe polynomial)
Define the following:
•
Polynomial: ___________________________________________________________________________
i.e.
non-example:
•
Terms: ____________________________________________________ i.e.
•
Coefficients: ________________________________________________ i.e.
•
Degree of a polynomial: _______________________________________ i.e.
•
Constant term: _______________________________________________ i.e.
•
Monomial: _____________________________________ i.e.
•
Binomial: ______________________________________ i.e.
•
Trinomial: ______________________________________ i.e.
Example 1) Which of these polynomials can be represented by the same algebra tiles?
a) 2 x 2  5 x  3
b) 5  3 p  2 p 2
c) 5m  3  2m 2
Example 2) Use algebra tiles to model each polynomial.
Is the polynomial a monomial, binomial, or trinomial? Explain.
a) 5x 2
b) 3b 2  5b  6
c) 4  2x
Example 3) Which polynomial does each group of algebra tiles represent?
Model A
Model B
Model C
Assignment P.214
Lesson 5.2: Like Terms and Unlike Terms (p.217)
Focus: Simplify polynomials by combining like terms
These are all zero pairs:
and
and
We can use zero pairs to simplify algebraic expressions.
Example 1) Combining Like Tiles and Removing Zero Pairs
Simplify this tile model. Write the polynomial that the remaining tiles represent.
a)
Polynomial ___________________________
b)
Polynomial ___________________________
c)
Polynomial ___________________________
Terms that can be represented by matching tiles are called __________ terms.
•
Examples of like terms: _______________________
__________________________
•
Examples of Unlike terms: _____________________
___________________________
We can simplify a ______________________ by adding the ______________________ of the _________ terms
Example 2) Simplifying a Polynomial Symbolically
Simplify:
a) 6a  5  2a  6
c) 19 y 2  9  5 x  5 y  x 2  4 y 2  2 x 2
b) 4 x 2  2  7 x  5 x  6 x 2  1  x
d) 7 xy  y 2  5x 2  3 yx  y  x  4 y 2  5x  y
5.3 Adding Polynomials (p.225)
Focus: Use different strategies to add polynomials
Investigate (p.225)
Example 1) Adding Polynomials with Algebra Tiles
o Use algebra tiles to model  3s 2  2s  6     s 2  4s  2  concretely.
o Draw the model pictorially below.
o Write an addition sentence using symbols (algebraically).
Pictorially
Symbolically
Example 2) Adding Polynomials Symbolically
Add:  5 x 2  7 x  8    7 x 2  16 x  5 
Example 3) Adding Polynomials Vertically
Add:  4a 2  3a  7    7 a 2  5a 
Example 4) Write a polynomial for the perimeter of this rectangle. Simplify the polynomial.
3x  7
2x  1
Substitute to check your answer.
Example 5) Adding Polynomials in Two Variables
 3a
2
 4a  5b  8ab  5b 2    6a 2  6ab  8b  9a 2 
5.4 Subtracting Polynomials (p.231)
Focus: Use different strategies to subtract polynomials.
Example 1) Represent the expression 5 x 2  2 x 2 in 3 different ways, then simplify.
Method 1: Using tiles (__________________)
Method 2: Using tiles (__________________)
Symbolically (algebraic)
Example 2) Represent the expression  4 x    3x  in 3 different ways, then simplify.
Method 1: Using tiles (__________________)
Method 2: Using tiles (__________________)
Symbolically (algebraic)
Example 3) Represent the expression  6 x 2    2 x 2  in 3 different ways, then simplify.
Method 1: Using tiles (__________________)
Symbolically (algebraic)
Method 2: Using tiles (__________________)
Example 4) Represent the expression  6 x  2    3x  4  in 3 different ways, then simplify.
Method 1: Using tiles (__________________)
Method 2: Using tiles (__________________)
Symbolically (algebraic)
Example 5) Represent the expression  3 x 2  2 x  7    2 x 2  3 x  2  in 3 different ways, then simplify.
Method 1: Using tiles (__________________)
Method 2: Using tiles (__________________)
Symbolically (algebraic)
Example 6) Simplify each algebraically.
a)  5x    3x   _____
d)
 3x 2  6 x  4    x 2  3x  2 
b)
 2 x 2  3x    4 x 2  6 x 
 ______________
 __________
c)
 3x 2  4 xy  y 2    5 x 2  6 xy  7 y 2 
e)
 3a 2  2a  1   a 2  4a  3
 ______________
 ______________________
5.5 Multiplying & Dividing a Polynomial by a Constant
FOCUS: Use different strategies to multiply & divide a polynomial by a constant
Discuss what 3 25 means? How can you represent this using a picture?
Tiles Applet
INVESTIGATE: (P.241)
A. 2  3x 
D. 2  3x 
B. 3  2 x  1
E. 3  2 x  1
C. 2  2 x 2  x  4 
F. 2  2 x 2  x  4 
G. 9x  3
J. 9 x   3
H.
8x  12  4
K.
I.
5x
8x  12   4
L.
5x
2
 10 x  20   5
2
 10 x  20    5 
Example 1) Multiplying a Binomial and a Trinomial by a Constant
Determine each product pictorially and symbolically:
a) 2  3m  4
Pictorially
Method 1
Method 2
Symbolically
Method 3
b) 3  n 2  2n  1
Pictorially
Symbolically
NOTE: Multiplication & division are ________________ operations.
To divide a polynomial by a constant, we ______________ the process of multiplication.
Representing division using a Model
 6x  3  3
Algebra Tiles
Area Model
Algebraically
Example 2) Dividing a Binomial and a Trinomial by a Constant
a)
b)
4 x 2  12
4
Method 1: Tiles
Method 2: Algebraic (breaking into terms)
3m2  15mn  21n 2
3
Method 1: Division = backwards multiplication
Method 2: Algebraic (breaking into terms)
5.6 Multiplying & Dividing a Polynomial by a Monomial (p.249)
FOCUS: use different strategies to multiply and divide a polynomial by a monomial
Example 1) Multiplying a Binomial by a Monomial
Represent each expression using tiles, then complete the product.
A)  2 x  4 x 
Method 1: Tiles
Method 2: Area model
Method 3: Algebraic
B)  3x  2 x  4
Method 1: Tiles
Method 2: Area model
Method 3: Algebraic
(using the distributive property)
C)  4c  2c  3
Method 1: Tiles
Method 2: Distributive property
Example 2)
A) 2a  5a 
B) 4b  3b  2 
C) 3c  5c  1
NOTE: As before, multiplication & division are ________________ operations.
To divide a polynomial by a monomial, we ______________ the process of multiplication.
6x2
Example 3) Draw
. Simplify.
2x
6 w2  9 w
Example 4) Draw
. Simplify.
3w
Example 5) Dividing a Monomial and a Binomial by a Monomial
Determine the quotient of each.
12m 2
36k  18k 2
A)
D)
3m
6k
B)
3g 2  9 g
3g
E)
18 f 2  12 f
C)
6f
24d 2  8d
4d