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



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



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

= =



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) _________________



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



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)



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



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



9



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 =



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



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! →



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



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



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



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



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



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



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



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



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



22

Extra practice:



23



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 =



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



26

MID-CHAPTER QUIZ: p.498



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



28



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



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:



31

Practice:

1) 2)

3) 4)

5) 6)

HW: p.508 #8-26 even, 34



32

Factoring ax2 + bx + c



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



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



35



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



37

Geometry Write a polynomial to express the area of each shaded region. Then write the polynomial in

factored form.



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



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



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



41



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Algebra 1 Mrs. Bondi Algebra 1 Unit 6: Polynomials · Algebra 1 Unit 6: Polynomials ... (PH Text...

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Transcript of Algebra 1 Mrs. Bondi Algebra 1 Unit 6: Polynomials · Algebra 1 Unit 6: Polynomials ... (PH Text...