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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Algebra 1
Unit 6: Polynomials
Lesson 1 (PH Text 7.1): Zero and Negative Exponents
Lesson 2 (PH Text7.2): Scientific Notation
Lesson 3 (PH Text 7.3): Multiplying Powers with the Same Base
Lesson 4 (PH Text 7.4): More Multiplication Properties of Exponents
Lesson 5 (PH Text 7.5): Division Properties of Exponents
Lesson 6 (PH Text 8.1): Adding and Subtracting Polynomials
Lesson 7 (PH Text 8.2): Multiplying a Polynomial by a Monomial
Lesson 8 (PH Text 8.2): Monomial Factors of Polynomials
Lesson 9 (PH Text 8.3): Multiplying Binomials
Lesson 10 (PH Text 8.4): Multiplying Polynomials: Special Cases
Lesson 11 (PH Text 8.5): Factoring x2 + bx + c, c > 0
Lesson 12 (PH Text 8.5): Factoring x2 + bx + c, c < 0
Lesson 13 (PH Text 8.6): Factoring ax2 + bx + c
Lesson 14 (PH Text 8.7): Factoring Special Cases
Lesson 15 (PH Text 8.8): Factoring by Grouping
Lesson 16 (PH Text 11.1): Simplifying Rational Expressions
1
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 1 (PH Text 7.1): Zero and Negative Exponents
Objective:
to simplify expressions involving zero and negative exponents
Properties:
Zero as an Exponent – For every nonzero number a, a0 = 1.
Examples:
80 =
(-4)0 =
(3.14)0 =
1
Negative Exponent – For every nonzero number a and integer n, a  n  n .
a
Examples:
8-2 =
(-4)-3 =
(3.14)-2 =
Discussion:
What about 00?
What about 9x0?
Class Practice:
1) 3-4 =
2) (7.89)0 =
6) 8x3y-2 =
8)
10)
1
4 3
4b
2 3
2 a
3) (2.5)-3 =
4) (-16)-2 =
7) 3-2x-9y5
9)
1
an
11)
HW: p.417 #8-58 even
2
7m0 n5
p 1
5) 2-1 =
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 2 (PH Text7.2): Scientific Notation
Objectives: to write numbers in scientific and standard notation
to compare and order numbers using scientific notation
Complete the table. Notice the pattern.
103
=
=
10 2
=
=
101
=
=
100
=
=
101
=
=
102
=
=
103
=
=
Scientific Notation – a number expressed in the form a x 10n , where n is an integer and 1 ≤ |a| < 10.
Examples:
1) Is the number written in scientific notation? Explain why/why not.
a) 2.36 x 104
b) 762.1 x 10-3
c) 0.41 x 10-8
2) Find each value.
a) 2.36 x 104 =
b) 7.1 x 10-3 =
Shortcut hint:
3) Write each number in scientific notation.
a) 18,459 =
b) 0.00987 =
Shortcut hint:
3
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Comparing Numbers in Scientific Notation:
First compare the powers of ten. If the numbers have the same power of 10, then compare the front parts.
Example: Write the numbers in order from least to greatest.
a) 1.23 x 107, 4.56 x 10-3, 7.89 x 103
b) 0.987 x 103, 654 x 103, 32.1 x 103
11. 3(4 x 105) _________________
12. 2(7 x 102) _________________
13. 10(8.2 x 1012) _________________
14. 6(3 x 108) _________________
HW: p.423 #9, 12-46 even
4
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 3 (PH Text 7.3): Multiplying Powers with the Same Base
Objectives: to identify monomials
to multiply monomials with the same base
Monomial - a real number, a variable, or a product of a real number and one or more variables
Examples:
1) Determine whether or not the given term is a monomial.
1
13
a) x 2 yz 2
b) 
c) 7g 23
2
2
8a 2 b 5
d)
c
Property: Multiplying Powers with the Same Base
When multiplying monomials with the same base, ADD the exponents:
a m  a n  a mn
a3  a5  (a  a  a)  (a  a  a  a  a)  a35 
2 3  2  2 3  21  2 31  2 4 
 Examples:

2) Write each expression using each base only once.
a) 23 · 25 · 2-4
b) (0.6)-9(0.6)-8
3) Simplify
a) a 2  b 3  a 2


d) 2 x 2 y 3x 5 y 2



b) 2 x  3x 3  4 x 2

g) 0.5 1013 0.3 104






e) 3m 2 n 5  8mn 2


c) 2r 3 x r 2 x d x
h) 4  10 6 3  10 3
HW: p. 429 #7, 9-63 multiples of three
5

f) 3103 7 109 

Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 4 (PH Text 7.4): More Multiplication Properties of Exponents
Objectives: to raise a power to a power
to raise a product to a power
Monomial(s) Raised to a Power:
When a monomial is raised to a power, you multiply the exponents
(a m )n  a mn
(ab)m  a mbm
Examples:
1) Simplify:
a)  (2 2 ) 3
b) [(2) 2 ]3
c) (2 2 ) 3

d) (x
6 9
)

e) (ab) 3
f) (8x5)3


j) (2a 2 ) 3 (3b) 2 
2
i) (-7 x 105)2

HW: p.436 #8-54 even
(GOOD IDEA: Mid-Chapter Quiz p.439)
6
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 5 (PH Text 7.5): Division Properties of Exponents
Objectives:
to divide monomials
to raise a quotient to a power
Property of Exponents for Division
When dividing monomials with the same base, SUBTRACT the exponents
am
 a mn
n
a
a 5 (a  a  a  a  a)

 a 53 
3
a
(a  a  a)
144 2  2  2  2  3 3 2 4  32

 3 2  2 43  322 
72
2  2  2  3 3
2 3

Examples:
1) Simplify:
a)

y6
y3
b)
y2
y7
c)
6  1012
f)
2  10 6
12k 2 m 3 n
e)
 9m 3 n 6 k 5
4a 2 b 5
d)
2a 6 b 2
y 16
y 16

6 x 3 y
2 2
g) 9 x y
2) Find the value of x in each equation:
3x
1
a) 9  3
3
3
Simplify:

 4 10 
 3 10 
6
e)
f)
3
9  10-12
-3  109
t 5 x
b) x  t 3
t
c 4
g)  
d 
  2c 
h)  2 
 d 
7
4
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
 (2a 2 )3 
i) 
2 
 (3a) 
2
 a 2 m3 
j)  m 1 
 a

2
HW: p.443 #8-52 even, 70, 80
If lessons 4 and 5 are combined…
HW: p.437 #22-36 even, 44-54 even; p.444 #21-24, 32-52 even, 70, 80
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 6 (PH Text 8.1): Adding and Subtracting Polynomials
Objectives: to classify, add, and subtract polynomials
Vocabulary:
Descending order - writing the order of the variables from highest power to lowest power
Ascending order - writing the order of the variables from lowest power to highest power
Monomial - has one term; example: 0.006t
Binomial - has two terms connected by addition or subtraction; example: 3x + 2
Trinomial - has three terms connected by addition or subtraction; example: 3x 2  2 x  1
Polynomial – is a monomial or a sum or difference of monomials
Degree of a term - exponent of the variable (each monomial is a term)
Degree of a polynomial – is the highest degree of any of its terms after it has been simplified
Polynomial
Degree
Name Using
Degree
Number
Of Terms
Name Using
Number of Terms
7x  4
3x 2  2 xy 2  1
4x 3 y4z
5
x3  x 2  x  1
Polynomials can be simplified by combining like terms.
Examples:
1) State the degree:
1
a) x
2
b) 8a 2 b 5
2 2
x y  8 xy 2  5
3
3) Simplify: 3r 2 s 2  5rs 3  9r 2 s 2  4s =
2) State the degree of 2 x 2 y 2 
4) Simplify: 0.3xy 2  0.9 x3 y  0.3xy 2  0.6 x3 y  2.4 =
10
c) 6
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Standard form
- terms are in alphabetical order
- terms decrease in degree from left to right - no terms have the same degree
(when more than one variable, with respect to the first variable in the alphabet)
Write each polynomial in standard form, then name each by its degree and number of terms
1)
2  7x
2)
2 x3  x 4  3
3)
3x 4  2  2 x 4  7 x
4)
6 x 2  7  9 x3
Some algebraic expressions are not polynomials
Polynomial
x 4  x 2  3
1
2
x 3
3
2 y  x2  z  8
3
x2
Why it is not a Polynomial
11
An algebraic expressions is NOT a
polynomial if it:
1)
has a negative exponent
2)
is not a sum or difference
3)
has a variable in the denominator
4)
has more than one variable
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Adding (two options):
1.
4 x 3  x 2  3x  5
x 2  4x  6
+
4 x3  x 2  3 x  5
vertical

b.
horizontal
4x3  x 2  3x  5  x 2  4x  6 
Subtracting (two options):
2.
x2  4x  6
a.
align like terms
BE CAREFUL!
 3

2
4a  3a  3a  5


–
 2

a  2a  7


4a3  3a 2  3a  5



a 2  2a  7
a.
vertical
b.
horizontal (add the opposite)
Distribute the negative! →
align like terms
 4a 3  3a 2  3a  5   a 2  2a  7  

 


 

3
2
2
4a  3a  3a  5  a  2a  7 
12
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
 3

8n  7n  4
3.


–
 3

2
3n  2n  7


8n3

a.
vertical
b.
horizontal

 7n  4
 3n 3  2n2
7
align like terms
Distribute the negative!
8n3  7n  4  3n3  2n 2  7 
HW: p.477 #9-27 multiples of 3, 30-40, 44-48
13
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 7 (PH Text 8.2): Multiplying a Polynomial by a Monomial
Objectives:
to multiply a polynomial by a monomial
to simplify algebraic expressions that involve multiplication of a polynomial by a monomial
Use the Distributive Property:
1.
5x2 y  3x 
2.
-6x(x2 – xy + y)
3.
 2a 2b 5ab2  3b  4a 7  
4.
4 x 2 2 x  1  3x6 x  4 
5.
6ab2  2a 2b  3a  5  3a 2b 4ab2  2b  






14
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
GCF and LCM with Variables
Objectives:
to find the greatest common factor and the least common multiple of a set of monomials
Greatest Common Factor (GCF)
Least Common Multiple (LCM)
Find the GCF of 32 and 24.
Method 1 – “Rainbow Method”
List all factors of 32 and 24.
32 – 1, 2, 4, 8, 16, 32
Method 2 – Prime Factorization
List the prime factors of 32 and 24.
32 – 25
24 – 1, 2, 3, 4, 6, 8, 12, 24
24 – 23·3
Common factors: 1, 2, 4, 8
common prime factor is 2
GCF = 8
lesser power of that prime factor is 23
GCF = 23 = 8
Method 3 – Ladder Method
2
32
24
Is there a common factor?
2
16
12
yes
2
8
6
yes
4
3
no
↑
GCF = 2·2·2 = 8
for LCM, “use the “L”
LCM = 2·2·2·4·3 = 96
Find the GCF of 36m3 and 45m8.
Method 1
List all factors of 36m3 and 45m8.
36m3 – 1, 2, 3, 4, 6, 9, 18, 36 · m· m· m
Method 2
List the prime factors of 36m3 and 45m8.
36m3 – 22·32· m3
45m8 – 1, 3, 5, 9, 15, 45 · m· m· m· m· m· m· m· m 45m8 – 32·5· m8
Common factors: 1, 3, 9
GCF = 9m3
· m· m· m
common prime factor is 3 and m
lesser power of that prime factor is 32 and m3
GCF = 32 · m3 = 9m3
15
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Find the GCF of 36m3 and 45m8 using the Ladder method.
3
36m3 45m8
Is there a common factor?
3
12 m3 15 m8
yes
m
4 m3
5 m8
yes
m
4 m2
5 m7
yes
m
4m
5 m6
yes
4
5 m5
no
↑
GCF = 3·3·m·m·m = 9m3
for LCM, “use the “L”
LCM = 3·3·m·m·m·4· 5m5 = 180m8
Practice:
Find the GCF.
1. 60x4 and 17x2 _______________
2. 32y12 and 36y8 _______________
3. 16n3 , 28n2 and 32n5 _______________
4. 16m10 , 18m and 30m3 _______________
Find the LCM.
5. 60x4 and 17x2 _______________
6. 32y12 and 36y8 _______________
7. 16n3 , 28n2 and 32n5 _______________
8. 16m10 , 18m and 30m3 _______________
Review: Multiply.
11. 4(x2 + 3x + 2) _______________
12. a(a + 7) _______________
13. 2p(p2 + 2p + 1) _______________
14. 3xy(z2 + 6z + 8) _______________
HW: p.482 #5-20
16
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 8 (PH Text 8.2): Monomial Factors of Polynomials
Objectives:
to factor the greatest common monomial factor from a polynomial
To factor a polynomial:
1 – Find the GCF of the terms.
2 – Use the distributive property (in reverse).
3 – … more to follow in future lessons ...
Sample:
1.
10xy – 15x2
← Find the GCF of 10xy and 15x2
5x(2y – 3x)
← Use the GCF, and what remains of each term
with the distributive property.
2.
 10a 3b3  6a 2b 2  8a 9b
3.
8x 3  14 x 2  12 x
4.
12a 2b 2  30ab2
17
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Geometry Application:
Reminder: Area of a Circle: The area of a circle is the product of  and the square of the radius.
A   r2
The rectangle has sides measuring 4 cm and 6cm. Find the area of the shaded
region.
HW: p.483 #21-28, 36
18
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 9 (PH Text 8.3): Multiplying Binomials
Objectives: to multiply two binomials, or a binomial and a trinomial
To multiply two binomials:
“Double” Distribute Method
1.
a  4a  5 
aa  5  4a  5
Table Method
(x – 7)(2x + 9)
Make a table of products.
2x
9
Write out all of the product terms and simplify.
x
-7
FOIL Method
(shortcut to other methods)
2.
s  6s  3 
3.
2x  3x  5 
F
O
I
L
–
–
–
–
First terms
Outer terms
Inner Terms
Last Terms
F
O
I
L
s  s  s  (3)  6  s  6  (3) 
F
O
I
L
2x  x  2x  (5)  3 x  3 (5) 
19
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
4.
4a  b2a  3b 
5.
8 j  32 
8 j  38 j  3
To multiply any two polynomials
6.
2t  5 3t  4t 2  2t 3 


“Double” Distribute Method (Horizontal)
a)
2t  3t  4t 2  2t 3   5 3t  4t 2  2t 3  




Arrange in descending order method (Vertical)
2t 3  4t 2  3t

2t  5
b)
20
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Multiply the polynomials:
7.
 a 2  3a  9 a  4




(Solve using both methods.)
“Double” Distribute Method (Horizontal)
a)
 a 2  3a  9  a   a 2  3a  9  4 








Arrange in descending order method (Vertical)
a 2  3a  9

a4
b)
HW: p.489 #13-14, 27-28, 30-42 even
21
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Extra practice:
22
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
23
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 10 (PH Text 8.4): Multiplying Polynomials: Special Cases
Objectives: To find the square of a binomial
To find the product of the sum and difference of two terms
Product of the sum and difference of same two terms:
m  nm  n = m2  mn  mn  n2 = m 2  n 2
STEPS
1. square the first term
 second term
2. square the
3. write the difference of the two squares

Examples:
1.
x  5x  5 
2.
 4x  y  4x  y  
Square of a binomial
m  n2 = m  nm  n = m2  mn  nm  n2 = m 2  2mn  n 2
or
m  n2 = m  nm  n = m2  mn  nm  n2 = m 2  2mn  n 2



STEPS
1. square the firstterm
2. double the product of the two terms
3. square the second term
4. write the sum of the three new terms

Examples:
3.
4x  2 y 2 
m  4x
n  2y
4.
3a  4b2 
m=
n=
24
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
2
 2x 2  3  
5.




6.
2
 5x 2  2  




7.
(23)2 = (20 + 3)2
8.
(41)2 = (
9.
 3x 2  y  3x 2  y  






)2
HW: p.496 #17, 26-52 even
25
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
MID-CHAPTER QUIZ: p.498
26
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 11 (PH Text 8.5): Factoring x2 + bx + c, c > 0
Objectives:
to factor a trinomial of the form x2 + bx + c, c > 0
Factoring:
Find two binomials that will multiply to be the quadratic expression given --- FOIL backwards.
1. Draw the parentheses.
(
)(
)
2. Put two first terms in the ( ) that will multiply to be the first term of the quadratic.
3. Find two second terms for the ( ) that will multiply to be the last term of the quadratic, but add to be
the middle term of the quadratic.
s  6s  3 
1)
x2 + 5x + 6
F
O
I
L
s  s  s  (3)  6  s  6  (3) 
2)
x2 – 13x + 12
3)
(x – 1)(x – __)
(x + 2)(x + __)
x2 – 18x + 17
(x – __)(x – __)
4)
x2 + 4x + 3
5)
x2 + 3x + 2
6)
x2 – 6x + 5
7)
11 – 12p + p2
8)
7 + 8m + m2
9)
d2 – 8d + 12
10)
21 – 10p + p2
11)
27 + 12x + x2
12)
d2 – 9d + 14
13)
x2 – 10x + 25
14)
x2 + 12x + 32
15)
x2 + 16x + 48
HW: p.503 #10-19
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
28
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 12 (PH Text 8.5): Factoring x2 + bx + c, c < 0
Objectives: to factor a trinomial of the form x2 + bx + c, c < 0
HW: p.503 #20-44 even
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 13 (PH Text 8.6): Factoring ax2 + bx + c
Objectives:
to factor a trinomial of the form ax2 + bx + c
The process of factoring a trinomial is finding two binomials whose product is the given trinomial.
Basically, we are reversing the FOIL method to get our factored form. We are looking for two binomials
that will result in the given trinomial when you multiplied.
Reverse FOIL Method - What you have been doing still works, but can get complicated with the leading
coefficient being something other than one.
try 8x2 + 10x – 3
Method 2
Example 1:
Step 1: Multiply the first and last terms
(6x)(-12x)=-72x2
Step 2: Find factors of -72 that will subtract or add to make +1 (coefficient of the middle term)
9x and -8x
Step 3: Replace the middle term with 9x and -8x
6x2 + 9x – 8x – 12
Step 4: Factor out the Greatest Common Factor from the 1st and 2nd terms and then from the 3rd and 4th
terms
6x2 + 9x – 8x – 12
3x(2x + 3) – 4(2x + 3)
works like 5a – 3a = (5 – 3)(a) = 2a
Step 5: Combine like terms (Final Answer)
(3x – 4)(2x + 3)
Step 6: Check to be sure it works … FOIL.
6x2 + 9x – 8x – 12 =

Example 2:
Example 3:
Step 1:
Step 1:
Step 2:
Step 2:
Step 3:
Step 3:
Step 4:
Step 4:
Step 5:
Step 5:
Step 6:
Step 6:
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Practice:
1)
2)
3)
4)
5)
6)
HW: p.508 #8-26 even, 34
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
2
Factoring ax + bx + c
32
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 14 (PH Text 8.7): Factoring Special Cases
Objectives:
to factor perfect square trinomials and the differences of two squares
A polynomial is considered completely factored when it is written as a product of prime polynomials, or one
that cannot be factored.
To factor a polynomial completely:
1 – Factor out the greatest monomial factor (GCF)
2 – If the polynomial has two or three terms, look for:
 A perfect square trinomial
 A difference of two squares
 A pair of binomial factors
3 – If there are four or more terms, group terms, if possible, in ways that can be factored. Then factor out
any common polynomials.
4 – Check that each factor is prime (cannot be factored any further).
5 – Check your answer by multiplying all the factors to be sure it returns to the original polynomial.
Examples:
1. 6x2 + 9x + 3
2. 20x3 – 28x2 + 8x
3. 5x4 – 50x3 + 125x2
Look for the following special cases:
Factor:
Difference of Two Squares
x 2  64
both terms are perfect squares
Perfect Square Trinomial
x 2  10 x  25
st
1 & 3rd terms are perfect squares
x 2  64
x 2  10 x  25
Examples:
4. 5x4 – 245x2
5. 2m3 – 36m2 + 162m
33
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Sometimes you may need to factor out the GCF before you can factor an expression into two binomials.
Factor :
10 x 2  40
Practice: Factor each expression.
1)
a 2  16
2)
x 2  49
3)
9 x 2  25
4)
25a 2  64
5)
18 x3  32 x
6)
4h3  36h
7)
4 x 2  12 x  9
8)
25c 2  40c  16
9)
16 x 2  24 x  9
HW: p.514 #10-42 even
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
35
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 15 (PH Text 8.8): Factoring by Grouping
Objectives:
to factor higher-degree polynomials by grouping
A polynomial is considered completely factored when it is written as a product of prime polynomials, or one
that cannot be factored.
To factor a polynomial completely:
1 – Factor out the greatest monomial factor (GCF)
2 – If the polynomial has two or three terms, look for:
 A perfect square trinomial
 A difference of two squares
 A pair of binomial factors
3 – If there are four or more terms, group terms, if possible, in ways that can be factored. Then factor out
any common polynomials.
4 – Check that each factor is prime.
5 – Check your answer by multiplying all the factors to be sure it returns to the original polynomial.
Examples:
5. 27x3 – 3xy2
6. 4m3 – 48m2 + 144m
7. 18x2 – 12x + 2
8. 8x2y3 + 4x2y2 – 12x2y
9. 2d5 – 162d
36
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Geometry Write a polynomial to express the area of each shaded region. Then write the polynomial in
factored form.
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
In the polynomial 6(a + b) + 3(a + b), the binomial (a + b) is common to both terms. The distributive
property can be used to factor out (a + b).
6(a + b) + 3(a + b)
Examples:
Factor.
5.
7(a + 2b) + (a + 2b) – 3(a + 2b)
6.
11(x – 3) + 7(3 – x)
7.
4d – 4 g + 9g – 9d
8.
25r – r3 – r2s + 25s
9.
49n2 – 9m2 + 24m – 16
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Directions for 20 and 21. A square is enclosed within another square. The area of the larger square is the
given polynomial; the area of the smaller is the monomial. Write a polynomial in factored form to represent
the difference of the two areas.
20.
a2 + ab + b2; 9b2
21.
4c2 + 72c + 324; 25c2
Reminder: Some polynomials may contain common binomial factors. Sometimes these binomial factors
are opposites, or additive inverses.
The additive inverse of a is –a.
Examples:
Are these polynomials additive inverses of each other?
1.
x – y and y – x
2.
2x + 1 and 2x – 1
3.
3t – 4 and 4 – 3t
4.
5y – 2 and 5y + 2
HW: p.519 #10-28 even, 35
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Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
Lesson 17 (PH Text 11.1): Simplifying Rational Expressions
Objectives:
to identify values for variables that make a rational expression undefined
to simplify rational expressions
Rational numbers are numbers that can be expressed as a fraction. The denominator cannot be zero.
A rational expression is similar, but usually contains two polynomials. The denominator still cannot be zero.
A rational expression is in its simplest form when the numerator and denominator have 1 as their only
common factor. The expression will have restrictions on the variable which will prevent the denominator
from being zero, called an excluded value.
Step1: Factor both the numerator and the denominator.
Step2: Find the restrictions on the denominator.
Step3: Simplify the expression.
Examples:
7
1.
a 3
4.
6.
Simplify and state the values for which each expression is undefined.
a3
12r
2.
3.
2
9a  1
20r 2
6a  4b
18
ah
ha
HW: p.655 #8-32 even, 43
40
5.
3 x
2x  5x  3
7.
m 2  2m  3
9  m2
2
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
41
Algebra 1
Mrs. Bondi
Unit 6 Notes: Polynomials
42