Algebra 1 Mrs. Bondi Algebra 1 Unit 6: Polynomials · Algebra 1 Unit 6: Polynomials ... (PH Text...
Transcript of Algebra 1 Mrs. Bondi Algebra 1 Unit 6: Polynomials · Algebra 1 Unit 6: Polynomials ... (PH Text...
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
1
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
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
2
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 =
Negative Exponent – For every nonzero number a and integer n, 1
n
naa
.
Examples: 8-2
= (-4)-3
= (3.14)-2
=
Discussion: What about 0
0?
What about 9x0?
Class Practice:
1) 3-4
= 2) (7.89)0 = 3) (2.5)
-3 = 4) (-16)
-2 = 5) 2
-1 =
6) 8x3y
-2 = 7) 3
-2x
-9y
5
8) 3
1
4 9)
1na
10) 2 3
4
2
b
a 11)
0 5
1
7m n
p
HW: p.417 #8-58 even
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
3
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.
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:
310
= =
210
= =
110
= =
010
= =
110
= =
210
= =
310
= =
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
4
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 10
3
b) 0.987 x 103, 654 x 10
3, 32.1 x 10
3
HW: p.423 #9, 12-46 even
11. 3(4 x 105) _________________ 12. 2(7 x 102) _________________ 13. 10(8.2 x 1012) _________________ 14. 6(3 x 108) _________________
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
5
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.
a) 22
2
1yzx b)
2
13 c) 237g d)
c
ba 528
Property: Multiplying Powers with the Same Base When multiplying monomials with the same base, ADD the exponents:
nmanama
23 2 23 21 231 24
a3 a5 (a a a) (a a a a a) a35
Examples:
2) Write each expression using each base only once.
a) 23 · 2
5 · 2
-4 b) (0.6)
-9(0.6)
-8
3) Simplify
a) 232 aba b) 23 432 xxx c) xxx drr 232
d) 252 32 yxyx e) 252 83 mnnm f)
3103 7109
g) 413 103.0105.0 h) 36 103104
HW: p. 429 #7, 9-63 multiples of three
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
6
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
( )
( )
m n mn
m m m
a a
ab a b
Examples:
1) Simplify:
a) 32 )2( b)
[(2)2]3 c)
(22)3
d)
(x6)9 e)
(ab)3 f) (8x
5)3
i) (-7 x 105)2 j)
(2a2)3(3b)2 2
HW: p.436 #8-54 even (GOOD IDEA: Mid-Chapter Quiz p.439)
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
7
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
nmana
ma
144
72
2 2 2 2 3 3
2 2 2 3 3
24 32
23 32 243 322
a5
a3
(a a a a a)
(a a a) a53
Examples:
1) Simplify:
a) 3
6
y
y b)
7
2
y
y c)
16
16
y
y
d) 26
52
2
4
ba
ba e)
563
32
9
12
knm
nmk
f)
6 1012
2 106
g) 22
3
9
6
yx
yx
e)
6
3
4 10
3 10
f)
-12
9
9 10
-3 10
2) Find the value of x in each equation:
a) 39 3
1
3
3
x
b) 35
tt
tx
x
Simplify: g)
c
d
4
h)
4
2
2
d
c
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
8
i)
22 3
2
( 2 )
(3 )
a
a
j)
2
1
32
m
m
a
a
HW: p.443 #8-52 even, 70, 80
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
9
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
10
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: 123 2 xx
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
7 4x
2 23 2 1x xy
34x y4z
5
3 2 1x x x
Polynomials can be simplified by combining like terms.
Examples:
1) State the degree:
a) x2
1 b) 528 ba c) 6
2) State the degree of 583
22 2222 xyyxyx
3) Simplify: 2 2 3 2 23 5 9 4r s rs r s s =
4) Simplify: 2 3 2 30.3 0.9 0.3 0.6 2.4xy x y xy x y =
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
11
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) 3 42 3x x
3) 4 43 2 2 7x x x 4) 32 976 xx
Some algebraic expressions are not polynomials
Polynomial Why it is not a Polynomial 4 2 3x x
2
1
3x
3 22 8y x z
2
3
x
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
12
Adding (two options):
1. 534 23 xxx + 642 xx
a. vertical
3 2
2
4 3 5
4 6
x x x
x x
align like terms
b. horizontal 64534 223 xxxxx
Subtracting (two options): BE CAREFUL!
2.
4a3 3a2 3a5
–
a2 2a 7
a. vertical
3 2
2
4 3 3 5
2 7
a a a
a a
align like terms
b. horizontal (add the opposite)
725334 223 aaaaa
725334 223 aaaaa
Distribute the negative! →
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
13
3.
8n3 7n 4
–
3n3 2n2 7
a. vertical
3
3 2
8 7 4
3n 2 7
n n
n
align like terms
b. horizontal Distribute the negative!
723478 233 nnnn
HW: p.477 #9-27 multiples of 3, 30-40, 44-48
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
14
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. xyx 325
2. -6x(x2 – xy + y)
3.
722 4352 ababba
4. 463124 2 xxxx
5.
babbaabaab 2435326 2222
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
15
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” Method 2 – Prime Factorization
List all factors of 32 and 24. List the prime factors of 32 and 24.
32 – 1, 2, 4, 8, 16, 32 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
↑ for LCM, “use the “L”
GCF = 2·2·2 = 8
LCM = 2·2·2·4·3 = 96
Find the GCF of 36m3 and 45m
8.
Method 1 Method 2
List all factors of 36m3 and 45m
8. List the prime factors of 36m
3 and 45m
8.
36m3
– 1, 2, 3, 4, 6, 9, 18, 36 · m· m· m 36m3
– 22·3
2· m
3
45m8 – 1, 3, 5, 9, 15, 45 · m· m· m· m· m· m· m· m 45m
8 – 3
2·5· m
8
Common factors: 1, 3, 9 · m· m· m common prime factor is 3 and m
GCF = 9m3 lesser power of that prime factor is 3
2 and m
3
GCF = 32
· m3
= 9m3
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
16
Find the GCF of 36m3 and 45m
8 using the Ladder method.
3 36m3 45m
8 Is there a common factor?
3 12 m3 15 m
8 yes
m 4 m3 5 m
8 yes
m 4 m2 5 m
7 yes
m 4 m 5 m6 yes
4 5 m5 no
↑ for LCM, “use the “L”
GCF = 3·3·m·m·m = 9m3
LCM = 3·3·m·m·m·4· 5m5
= 180m8
Practice: Find the GCF.
1. 60x4 and 17x
2 _______________ 2. 32y
12 and 36y
8 _______________
3. 16n3 , 28n
2 and 32n
5 _______________ 4. 16m
10 , 18m and 30m
3 _______________
Find the LCM.
5. 60x4 and 17x
2 _______________ 6. 32y
12 and 36y
8 _______________
7. 16n3 , 28n
2 and 32n
5 _______________ 8. 16m
10 , 18m and 30m
3 _______________
Review: Multiply.
11. 4(x2 + 3x
+ 2) _______________ 12. a(a
+ 7) _______________
13. 2p(p2 + 2p
+ 1) _______________ 14. 3xy(z
2 + 6z
+ 8) _______________
HW: p.482 #5-20
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
17
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 15x
2
5x(2y – 3x) ← Use the GCF, and what remains of each term
with the distributive property.
2. bababa 92233 8610
3. xxx 12148 23
4. 222 3012 abba
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
18
Geometry Application: Reminder: Area of a Circle: The area of a circle is the product of and the square of the radius.
2r A
The rectangle has sides measuring 4 cm and 6cm. Find the area of the shaded
region.
HW: p.483 #21-28, 36
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
19
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. 54 aa 545 aaa
Table Method (x – 7)(2x + 9) Write out all of the product terms and simplify.
Make a table of products.
FOIL Method F – First terms
(shortcut to other methods) O – Outer terms
I – Inner Terms
L – Last Terms
2. 36 ss F O I L
( 3) 6 6 ( 3)s s s s
3. 532 xx F O I L
2 2 ( 5) 3 3 ( 5)x x x x
2x 9
x
-7
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
20
4. baba 324
5. 238 j 3838 jj
To multiply any two polynomials
6.
32 24352 tttt
“Double” Distribute Method (Horizontal)
a)
3232 24352432 ttttttt
Arrange in descending order method (Vertical)
b)
52
342 23
t
ttt
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
21
Multiply the polynomials:
7. 4932
aaa (Solve using both methods.)
“Double” Distribute Method (Horizontal)
a) 2 23 9 3 9 4a a a a a
Arrange in descending order method (Vertical)
b)
2 3 9
4
a a
a
HW: p.489 #13-14, 27-28, 30-42 even
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
22
Extra practice:
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
23
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
24
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:
mn mn =
m2 mnmnn2 = 22 nm
STEPS
1. square the first term
2. square the second term
3. write the difference of the two squares
Examples:
1. 55 xx
2. 4 4x y x y
Square of a binomial
mn 2 =
mn mn =
m2 mnnmn2 = 22 2 nmnm
or
mn 2 =
mn mn =
m2 mnnmn2 = 22 2 nmnm
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. 224 yx
xm 4 yn 2
4. 243 ba
m =
n =
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
25
5.
2
2 32x
6.
2
2 25x
7. (23)2 = (20 + 3)
2
8. (41)2 = ( )
2
9. 2 23 3x y x y
HW: p.496 #17, 26-52 even
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
26
MID-CHAPTER QUIZ: p.498
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
27
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.
36 ss F O I L
( 3) 6 6 ( 3)s s s s
1) x2 + 5x + 6 2) x
2 – 13x + 12 3) x
2 – 18x + 17
(x + 2)(x + __) (x – 1)(x – __) (x – __)(x – __)
4) x2 + 4x + 3 5) x
2 + 3x + 2 6) x
2 – 6x + 5
7) 11 – 12p + p2 8) 7 + 8m + m
2 9) d
2 – 8d + 12
10) 21 – 10p + p2 11) 27 + 12x + x
2 12) d
2 – 9d + 14
13) x2 – 10x + 25 14) x
2 + 12x + 32 15) x
2 + 16x + 48
HW: p.503 #10-19
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
28
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
29
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
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
30
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 2
nd terms and then from the 3
rd and 4
th
terms
6x2 + 9x – 8x – 12
3x(2x + 3) – 4(2x + 3)
Step 5: Combine like terms (Final Answer) works like 5a – 3a = (5 – 3)(a) = 2a
(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:
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
31
Practice:
1) 2)
3) 4)
5) 6)
HW: p.508 #8-26 even, 34
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
32
Factoring ax2 + bx + c
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
33
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 – 28x
2 + 8x
3. 5x4 – 50x
3 + 125x
2
Look for the following special cases: Difference of Two Squares Perfect Square Trinomial
642 x 25102 xx
both terms are perfect squares 1st & 3rd terms are perfect squares
Factor: 642 x 25102 xx
Examples:
4. 5x4 – 245x
2 5. 2m
3 – 36m
2 + 162m
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
34
Sometimes you may need to factor out the GCF before you can factor an expression into two binomials.
Factor : 4010 2 x
Practice: Factor each expression.
1) 2 16a 2) 2 49x 3) 29 25x
4) 225 64a 5) 318 32x x 6) 34 36h h
7) 24 12 9x x 8) 225 40 16c c 9) 216 24 9x x
HW: p.514 #10-42 even
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
35
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
36
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 – 3xy
2
6. 4m3 – 48m
2 + 144m
7. 18x2 – 12x + 2
8. 8x2y
3 + 4x
2y
2 – 12x
2y
9. 2d5 – 162d
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
37
Geometry Write a polynomial to express the area of each shaded region. Then write the polynomial in
factored form.
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
38
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 – r
2s + 25s
9. 49n2 – 9m
2 + 24m – 16
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
39
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 + b
2; 9b
2 21. 4c
2 + 72c + 324; 25c
2
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
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
40
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: Simplify and state the values for which each expression is undefined.
1. 7
3a 2.
2
3
9 1
a
a
3.
2
12
20
r
r
4.
6 4
18
a b
5. 2
3
2 5 3
x
x x
6. a h
h a
7.
2
2
2 3
9
m m
m
HW: p.655 #8-32 even, 43
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
41
Algebra 1 Mrs. Bondi
Unit 6 Notes: Polynomials
42