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UNIT SEVEN: POLYNOMINALS

Suggested time: ~ 4 weeks
Targeted finish date: April 23

Section 7.1: Adding and Subtracting Polynomials                                       pp. 285 – 289
2 – 3 periods

Terms:

Monomial -       an algebraic expression with one term.      Example: 7, 3x, -4xy2.

Binomial -       an algebraic expression with two terms.     Example: 7x +5, 14x2 – 3x.

Trinomial -      an algebraic expression with three terms.   Example: 15x2 – 7x + 5.

Polynomial - an algebraic expression with one or more terms.

Like Terms - have the same variable raised to the same power.
Examples: 6x and –4x; 5xy2 and 2xy2; 20a 2b3c 4 and 5a 2b3c 4

Examples:

1.       Simplify:
(2 x 2  3x  4)  ( x 2  2 x  5)
 (2 x 2  x 2 )  (3x  2 x)  (4  5)
 3x 2  1x  1

2.       Simplify and evaluate at x = 2.
(2 x 2  3x  5)  ( x 2  x  1)
  x2  2 x  4

At x = 2:
 x2  2 x  4
 (2) 2  2(2)  4
 4  4  4
 4

Grade 9 Math                                          -1-                 Unit 7: Polynomials
2. Subtracting Polynomials
To subtract polynomials, add the opposite and combine like terms.

Examples:

1.     Write the opposite of each:

a.         7x2                        7x 2
b.         6 x3  5                    6 x3  5
c.         4 x 2  6 xy  4 y          4 x 2  6 xy  4 y

2.     Simplify:
(2 x  1)  ( 4 x  2)
 (2 x  1)  (4 x  2)
 6x 1

3.     Simplify and evaluate at x = 2 and x = -1:
(3 x 2  2 x  5)  (2 x 2  x  6)
 (3 x 2  2 x  5)  (2 x 2  x  6)
 x 2  3 x  11

At x = 2:                                      At x = -1:
x 2  3 x  11                                 x 2  3 x  11
 (2) 2  3(2)  11                            (1) 2  3(1)  11
 4  6  11                                   1  3  11
 1                                           13

Remember: Order of Operations – BEDMAS

3(2) 2  3(4)  12           NOT 3(2) 2  (6) 2  36

12  (1)(1)  1           VS.    (1) 2  (1)(1)  1

Assign:          p. 287, #1, 5, 8, 9, 11, 12, 14 – 18.

Grade 9 Math                                        -2-                                 Unit 7: Polynomials
Review of Algebra Tiles

Algebra Tiles are based on area.

Short side = 1
Long side = x

1 x 1 = +1                                         1 x 1 = -1

1 x x = +x                                         1 x x = -x

x x x = +x2                                     x x x = -x2

DARK = +                                          LIGHT = -

Examples:

1.     Use tiles to represent 2 x 2  3x  4 .

2.     What expression is shown?          x2  4 x  2

3.     Determine the perimeter and area of the following algebra-tiles.

A  lw
P  4x  8              x( x  4)
 x2  4x

Grade 9 Math                                     -3-                           Unit 7: Polynomials
4.     Add using algebra tiles:        ( x 2  2 x  1)  ( x 2  3 x  3)

+

=

=       2 x2  x  2

NOTE: Zero Property:            (+1) + (-1) = 0                +        =0

5.     Subtract using algebra tiles:   ( x 2  2 x  1)  (2 x 2  1x  3)

NOTE: To subtract with algebra tiles, add the opposite (FLIP the tiles).

_

_

=

=        x2  x  4

Assign:        p. 283, #1 – 3.
p. 287, #1, 3, 4, 7.

Grade 9 Math                                  -4-                            Unit 7: Polynomials
Section 7.2: Multiplying and Dividing Polynomials                                  pp. 290 – 294
2 periods
1. Multiplying Monomials:
To multiply two monomials, multiply their coefficients and multiply their variables (by

Recall the meaning of exponents,

(3x 2 )(5 x3 )  (3  x  x)(5  x  x  x)  15 x5

Examples:

1.     Simplify each of the following:
a.     2 x(3x)  6 x 2
b.     3a(4ab)  12a 2b
c.     2 x 2 (3x 2 y )  6 x 4 y
d.     3abc(4a 2b2 c)  12a3b3c 2
e.     (5 y )(3 y)(2 z )  30 y 2 z
f.     (2ab)(2ab)(3ab 2 )  12a 3b 4

2. Dividing Monomials:
To divide two monomials, divide their coefficients and divide their variables (by
subtracting their exponents).

Recall the meaning of exponents,

18 x 6 18  x  x  x  x  x  x
                            6 x4
3x 2
3 x  x

Examples:

2.     Simplify each of the following:
x7
a.        4
 x3
x
15a
b.               3a
5
18ab
c.                 3b
6a
9 x 4 y 3
d.           5
 3 x 9 y 2
3x y
15 x 6 y 7                5x2
e.                      5 x 2 y 2  2
3x 4 y 9                   y

Grade 9 Math                                           -5-               Unit 7: Polynomials
Sometimes we multiply monomials that are powers.

3. Powers of Monomials:

Recall the law of exponent for powers of powers – multiply exponents.

Examples:

3.     Simplify each of the following:
a.     ( x 2 y 3 )2  x 4 y 6
b.     (2 x 2 y 3 z 4 )3  8 x 6 y 9 z12
c.     (2a 2b3 ) 2  4a 4b 6
d.     (2a 2b3 )3  8a 6b9
e.     (2a 2b3 ) 4  16a8b12
f.     (3x 4 ) 2 (2 x)3  (9 x8 )(8 x3 )  72 x11

Assign:        p. 292, #1 – 3, 5 – 9, 11, 13 – 18.

Grade 9 Math                                     -6-                    Unit 7: Polynomials
Section 7.3: Multiplying a Polynomial by a Monomial                                        pp. 295 – 299
2 periods

To multiply a polynomial by a monomial, we use the distributive property. When we do this, we
say we are expanding the product.

Examples:

1.        Expand each of the following:

a.       3x(3x  2)  9 x 2  6 x

b.       3 y (2 y 2  3 y  2)  6 y 3  9 y 2  6 y

c.
3x(4 x  2)  2 x(5 x  6)
 12 x 2  6 x  10 x 2  12 x
 22 x 2  6 x

d.
4 x(2 x  1)  3x(5 x  4)
 8 x 2  4 x  15 x 2  12 x
 7 x 2  16 x

2.        Determine the product using algebra tiles: 2 x(3x  1)  6 x 2  2 x

Assign:            p. 298, #1, 4, 3 – 8 (a, c, e, g).

Grade 9 Math                                         -7-                         Unit 7: Polynomials
Section 7.6: Multiplying Two Binomials                                                    pp. 309 – 313
2 – 3 periods

To multiply two binomials, we use the distributive property twice. That is, we multiply
each term of one binomial by each term of the other binomial. This is also known as writing the
polynomial in expanded form.

Multiply:      ( x  2)( x  3)

O
F
( x  2)( x  3)
I
L
 x( x  3)  2( x  3)
 x( x)  x(3)  2( x)  2(3)
 x 2  3x  2 x  6
 x2  5x  6

This is also known as the FOIL method, where:

F = First
O = Outside
I = Inside
L = Last

Examples:

1.     Multiply:        (2 x  4)(3x  1)                     OR    “Magic Square”

O                                                          3x        -1
F                                                       2x   6x2       -2x
(2 x  4)(3x  1)                                           +4   12x        -4
I
L
 2 x(3x  1)  4(3 x  1)
 2 x(3x)  2 x(1)  4(3 x)  4( 1)
 6 x 2  2 x  12 x  4
 6 x 2  10 x  4

Grade 9 Math                                       -8-                          Unit 7: Polynomials
2.     Expand:          (3x  2)(5 x  4)

O
F
(3x  2)(5 x  4)
I
L
 3x(5 x  4)  2(5 x  4)
 3x(5 x)  3x(4)  2(5 x)  2(4)
 15 x 2  12 x  10 x  8
 15 x 2  22 x  8

3.     Multiply:        ( x  5)( x  5)

O
F
( x  5)( x  5)
I
L
 x( x  5)  5( x  5)
 x( x)  x(5)  5( x)  5(5)
 x 2  5 x  5 x  25
 x 2  25

NOTE:          Notice in this example the middle terms cancel out each other.
This is called a difference of squares. Do you know why?

4.     Expand:          ( x  5) 2           Be careful! This means ( x  5)( x  5) !

O
F
( x  5)( x  5)
I
L
 x( x  5)  5( x  5)
 x( x)  x(5)  5( x)  5(5)
 x 2  5 x  5 x  25
 x 2  10 x  25

Grade 9 Math                                        -9-                           Unit 7: Polynomials
5.     Expand:          ( x  5) 2           Be careful! This means ( x  5)( x  5) !

O
F
( x  5)( x  5)
I
L
 x( x  5)  5( x  5)
 x( x)  x(5)  5( x)  5(5)
 x 2  5 x  5 x  25
 x 2  10 x  25

6.     Determine the product using algebra tiles: (2 x  1)( x  3)  2 x 2  6 x  1x  3  2 x 2  7 x  3

Multiplying two binomials using algebra tiles is very similar to multiplying a monomial
by a binomial using algebra tiles.

Assign:        p. 312, #8 – 11, 13, 15 – 17.

Grade 9 Math                                       - 10 -                         Unit 7: Polynomials
Perimeter and Area Problems:

Examples:

1.     Determine the perimeter of the figure below.

x                                                P  4( x)  4( x  2)  4(3 x)
P  4 x  4 x  8  12 x
x+2                                                  P  20 x  8
3x

2.     Determine the perimeter and area of the figure below.

2x + 1

A          4x

6x + 3
B               4x + 2

(6x + 1)

Perimeter:
P  (6 x  3)  2(2 x  1)  4 x  (4 x  2)  (6 x  1)
P  (6 x  3)  4 x  2  4 x  (4 x  2)  (6 x  1)
P  24 x  8

Area:
Total Area = Area(A) +Area (B)

A  (2 x  1)(2 x  1)  (4 x  2)(6 x  1)
A  (4 x 2  2 x  2 x  1)  (24 x 2  4 x  12 x  2)
A  28 x 2  20 x  3

3.     Determine the perimeter and area of the figure below.

2x + 1                     Perimeter = 20x - 2

3x                                  Area = 12 x2  18x  12

x–4

Grade 9 Math                                         - 11 -                            Unit 7: Polynomials
Section 7.4 and 7.7: Factoring Polynomials                          pp. 300 – 305; 314 – 320
2 periods + 3 – 4 periods

In the previous sections, we have multiplied polynomials to write them in expanded form.
This section will focus on the inverse (or reverse) operation, that is, writing polynomials from
expanded form to FACTORED FORM.

Factored Form vs. Expanded Form:

   ( x  1)( x  2)  x 2  3x  2
(factor)(factor) = Product

Factored Form  Expanded Form

   x 2  3x  2  ( x  1)( x  2)
Product = (factor)(factor)
Expanded Form  Factored Form

There are FOUR main types of factoring:
1. Factoring out a Greatest Common Factor (GCF)
2. Factoring a Difference of Squares
3. Factoring a Trinomial ax 2  bx  c where a  1 (Product/Sum Method)
4. Factoring a Trinomial ax 2  bx  c where a  1 (Trial/Error Method or Decomposition)

1. Factoring out a Greatest Common Factor (GCF)

Greatest Common Factor
To determine the greatest common factor of two or more numbers or expressions, first of
all, write each as a product of its prime factors.
The greatest common factor is the product of the common prime factors.

Examples:

1.     Determine the GCF for each of the following:

a.     24 and 36

24                                  36
=4x6                                =6x6
=2x2x2x3                            =2x3x2x3

GCF      = all the factors that are common between the two
= 2 x 2 x 3 = 12

Grade 9 Math                                      - 12 -                 Unit 7: Polynomials
b.      3x2 and 12x .                                  c.         21x 3 y 2 and 14x 2 y 3 .

3x 2  3 x x                                              21x3 y 2  3 7 x x x y y
12x  2 2 3 x                                              14 x 2 y 3  2 7 x x y y y

Therefore, GCF = 3x                                        Therefore, GCF = 7x 2 y 2

d.        4a3b 2c 4 d 5 and 8a 6b7 c 2 d 3 .

GCF = 4a3b 2c 2 d 3

To factor out a GCF:
1. Determine the GCF
2. Write each term as a product of the GCF and another monomial
3. Use the distributive law to write the sum as a product.

2.     Factor each of the following polynomials by removing a GCF:

a.        5x  25                                      b.         9 x 2 y 2  6 xy 2

GCF = 5                                                    GCF = 3xy 2
5 x  35                                                9 x 2 y 2  6 xy 2
 5( x)  5(7)                                           3 xy 2 (3 x)  3 xy 2 (2)
 5( x  7)                                              3 xy 2 (3 x  2)

c.        3x  12 x 2                                  d.         16 x 4  6 x3  14 x 2

GCF = 3x                                                   GCF = 2x 2
3x  12 x 2                                             16 x 4  6 x 3  14 x 2
 3x(1)  3x(4 x)                                        2 x 2 (8 x 2 )  2 x 2 (3 x)  2 x 2 ( 7)
 3x(1  4 x)                                            2 x 2 (8 x 2  3 x  7)

3.     Simplify, then factor: 5x2  2 x  3  2 x 2  19 x  4 .

5 x 2  2 x  3  2 x 2  19 x  4
 7 x 2  21x  7
 7( x 2  3 x  1)

Assign:              p. 304, # 2 – 13 (a, c, e, . . . in each).

Grade 9 Math                                        - 13 -                                    Unit 7: Polynomials
2. Factoring a Difference of Squares

In an earlier section, when we multiplied a binomial by its conjugate the middle terms
cancelled each other out resulting in a binomial called a difference of squares.

( x  5)( x  5)
 x( x  5)  5( x  5)
 x( x)  x( 5)  5( x)  5( 5)
 x 2  5 x  5 x  25
 x 2  25

A difference of squares is a polynomial that can be expressed as x 2  y 2 .

Factoring a difference of squares  x 2  y 2  ( x  y )( x  y )

Examples:

1.     Factor the following:

a.      x 2  25                    b.       x2  9               c.        9 x2  16

x 2  25                             x2  9                         9 x 2  16
 x 2  52                            x 2  32                      (3 x) 2  42
 ( x  5)( x  5)                    ( x  3)( x  3)              (3 x  4)(3 x  4)

Be Careful of some Common Mistakes!

d.      x 2  25

This binomial cannot be factored. It is not a difference of squares but rather a sum of
squares!

e.      x 2  25 x  x( x  25)

This is not a difference of squares but rather there is a common factor!

f.      3x2  12                    NOTE: There is a common factor that can be removed first!

3x 2  12
 3( x 2  4)
 3( x  2)( x  2)

Grade 9 Math                                       - 14 -                         Unit 7: Polynomials
3. Factoring a Trinomial ax 2  bx  c where a  1 (Product/Sum Method)

To factor trinomials of the form ax 2  bx  c , find two numbers whose product is c and
whose sum is b. This is known as the Product/Sum Method.

To factor x 2  7 x  12 , for instance, we need a product of 12 and a sum of 7.

We list all the factors of 12:

Factors of 12               Sum
1, 12                1 + 12 = 13
2, 6                 2+6=8
3, 4                 3+4=7
-1, -12           (-1) + (-12) = -13
-2, -6             (-2) + (-6) = -8
-3, -4             (-3) + (-4) = -7

As we can see from the table, the two numbers that give a product of 12 and a sum of 7
are 3 and 4. Therefore, the factors are (x + 3) and (x + 4).

That is, x 2  7 x  12  ( x  3)( x  4) .       NOTE: Can check using FOIL!

Examples:

1.     Factor each of the following trinomials:

a.      x2  5x  6                Need a Product = 6 and Sum = 5           2 and 3

x 2  5 x  6  ( x  2)( x  3)

b.      x2  8x  15               Need a Product = 15 and Sum = -8         -3 and -5

x 2  8 x  15  ( x  3)( x  5)

c.      x2  x  6                 Need a Product = -6 and Sum = -1         2 and -3

Grade 9 Math                                     - 15 -                    Unit 7: Polynomials
x 2  x  6  ( x  2)( x  3)

d.        x 2  3x  28              Need a Product = -3 and Sum = -28              -4 and 7

x 2  3x  28  ( x  4)( x  7)

Sometimes you may need to use more than one method when factoring.

2.     Factor each of the following by removing a common factor first:

a.        5x2  10 x  40                       Remove common factor of 5 first.

5 x 2  10 x  40
 5( x 2  2 x  8)                   Need a Product = -8 and Sum = -2            -4 and 2
 5( x  4)( x  2)

b.        4 x 2  16 x  20                     Remove common factor of 4 first.

4 x 2  16 x  20
 4( x 2  4 x  5)                   Need a Product = -5 and Sum = 4             -1 and 5
 4( x  1)( x  5)

3.     Simplify, then factor: 5x  3x 2  15  4 x 2  3  3x

5 x  3 x 2  15  4 x 2  3  3 x
 x 2  8 x  12                      Need a Product = 12 and Sum = -8            -2 and -6
 ( x  2)( x  6)

Assign:           p. 318, #3, 7, 9, 11 – 16.

4. Factoring a Trinomial ax 2  bx  c where a  1 (Trial/Error Method or Decomposition)

This method of factoring will not be addressed until next year.

Grade 9 Math                                        - 16 -                         Unit 7: Polynomials
Section 7.5: Diving a Polynomial by a Monomial                                   pp. 306 – 308
2 periods

To divide a polynomial by a monomial, one of two methods may be used.

1. Divide each term of the polynomial by the monomial, or
2. Factor the polynomial and then divide the monomial into the common factor.

Examples:

1.     Simplify each of the following:

4 x2  8x
a.
2x

Method 1:                                  Method 2:

4 x2  8x                                4 x2  8x
2x                                       2x
4 x2 8x                                  4 x( x  2)
                                       
2x 2x                                       2x
 2x  4                                  2( x  2)  2 x  4

3b3  6b 2  9b
b.
6b

Method 1:                                  Method 2:

3b3  6b 2  9b
3b  6b  9b
3     2
6b
6b                                   3b(b 2  2b  3)

3
3b 6b 2 9b                                      6b
         
6b 6b 6b                                 b  2b  3
2

1        3                             
 b2  b                                       2
2        2                               1           3
 b2  b 
2           2

Grade 9 Math                             - 17 -                        Unit 7: Polynomials
10 x 2 y  15 xy 2  5 xy
c.
5 xy

Method 1:                                      Method 2:

10 x 2  15 xy 2  5 xy                  10 x 2  15 xy 2  5 xy
5 xy                                     5 xy
10 x 2 15 xy 2 5 xy                      5 xy (2 x  3 y  1)
                                      
5 xy    5 xy 5 xy                                5 xy
 2x  3 y 1                             2x  3y 1

Assign:            p. 307, #1, 2, 4 - 6.

Review:            p. 324

Self-Test:         p. 326

TEST

Grade 9 Math                                 - 18 -                         Unit 7: Polynomials

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