Lesson 1 (PH Text 7.1): Zero and Negative Exponents

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

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, .

Examples: 8-2 = (-4)-3 = (3.14)-2 =

Discussion:

What about 00?

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-9y5

8) 9)

10) 11)

HW: p.417 #8-58 even

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 ax 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 10-3c) 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:

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

HW: p.423#9, 12-46 even

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) b) c) d)

Property: Multiplying Powers with the Same Base

When multiplying monomials with the same base, ADD the exponents:

Examples:

2)Write each expression using each base only once.

a)23 · 25 · 2-4b) (0.6)-9(0.6)-8

3)Simplify

a)b) c)

d)e)f)

g)h)

HW: p.429 #7, 9-63 multiples of three

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

Examples:

1) Simplify:

a) b) c)

d)e)f) (8x5)3

i) (-7 x 105)2j)

HW: p.436#8-54 even (GOOD IDEA: Mid-Chapter Quiz p.439)

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

Examples:

1) Simplify:

a) b) c)

d) e) f)

g) e) f)

2) Find the value of x in each equation:

a) b)

Simplify:g)h)

i)j)

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

Lesson 6 (PH Text 8.1): Adding and Subtracting Polynomials

Objectives: toclassify, 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:

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
y4z
5

Polynomials can be simplified by combining like terms.

Examples:

1)State the degree:

a) b) c) 6

2)State the degree of

3)Simplify: =

4)Simplify: =

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)

3)4)

Some algebraic expressions are not polynomials

Polynomial / Why it is not a Polynomial

Adding (two options):

1. +

a.verticalalign like terms

b.horizontal

Subtracting (two options):BE CAREFUL!

2.–

a.verticalalign like terms

b.horizontal (add the opposite)

3.–

a.verticalalign like terms

b.horizontalDistribute the negative!

HW: p.477#9-27 multiples of 3, 30-40, 44-48
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:

2.-6x(x2 – xy + y)

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, 3232 – 25

24 – 1, 2, 3, 4, 6, 8, 12, 2424 – 23·3

Common factors: 1, 2, 4, 8common prime factor is 2

GCF = 8lesser power of that prime factor is 23

GCF = 23 = 8

Method 3 – Ladder Method

23224Is there a common factor?

21612yes

286yes

43no

↑for LCM, “use the “L”

GCF = 2·2·2 = 8LCM = 2·2·2·4·3 = 96

Find the GCF of 36m3 and 45m8.

Method 1Method 2

List all factors of 36m3 and 45m8.List the prime factors of 36m3 and 45m8.

36m3– 1, 2, 3, 4, 6, 9, 18, 36 ·m·m·m36m3 – 22·32·m3

45m8 – 1, 3, 5, 9, 15, 45 ·m·m·m·m·m·m·m·m45m8 – 32·5·m8

Common factors: 1, 3, 9 ·m·m·mcommon prime factor is 3 and m

GCF = 9m3lesser power of that prime factor is 32 and m3

GCF = 32 ·m3 = 9m3

Find the GCF of 36m3 and 45m8 using the Ladder method.

336m345m8Is there a common factor?

312m315m8yes

m4m35m8yes

m4m25m7yes

m4m5m6yes

45m5no

↑for LCM, “use the “L”

GCF = 3·3·m·m·m = 9m3LCM = 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

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.

Geometry Application:

Reminder: Area of a Circle:The area of a circle is the product of and the square of the radius.

The rectangle has sides measuring 4 cm and 6cm. Find the area of the shaded region.

HW: p.483#21-28, 36

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

Table Method

(x – 7)(2x + 9) Write out all of the product terms and simplify.

Make a table of products.

FOIL MethodF– First terms

(shortcut to other methods)O– Outer terms

I– Inner Terms

L– Last Terms

2. F O I L

3. F O I L

To multiply any two polynomials

“Double” Distribute Method (Horizontal)

Arrange in descending order method (Vertical)

Multiply the polynomials:

7.(Solve using both methods.)

“Double” Distribute Method (Horizontal)

Arrange in descending order method (Vertical)

HW: p.489#13-14, 27-28, 30-42 even

Extra practice:

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:

= =

STEPS

square the first term
square the second term
write the difference of the two squares

Examples:

Square of a binomial

= = =

STEPS

square the first term
double the product of the two terms
square the second term
write the sum of the three new terms

Examples:

m =

n =

7.(23)2 = (20 + 3)2

8.(41)2 = ( )2

HW: p.496#17, 26-52 even

MID-CHAPTER QUIZ: p.498
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.

Draw the parentheses.( )( )
Put two first termsin the ( ) that will multiply to be the first term of the quadratic.
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.

F O I L

1)x2 + 5x + 62)x2 – 13x + 123)x2 – 18x + 17

(x + 2)(x + __)(x – 1)(x – __)(x – __)(x – __)

4)x2 + 4x + 35)x2 + 3x + 26)x2 – 6x + 5

7)11 – 12p + p28)7 + 8m + m29)d2 – 8d + 12

10)21 – 10p + p211)27 + 12x + x212)d2 – 9d + 14

13)x2 – 10x + 2514)x2 + 12x + 3215)x2 + 16x + 48

HW: p.503#10-19

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

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

Method2

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)

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:

Practice:

1) 2)

3) 4)

5) 6)

HW: p.508#8-26 even, 34
Factoring ax2 + bx + c

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:

6x2+9x + 3
20x3 – 28x2 + 8x
5x4 – 50x3 + 125x2

Look for the following special cases: Difference of Two Squares Perfect Square Trinomial

both terms are perfect squares 1st & 3rd terms are perfect squares

Factor:

Examples:

5x4 – 245x25. 2m3 – 36m2 + 162m

Sometimes you may need to factor out the GCF before you can factor an expression into two binomials.

Factor :

Practice: Factor each expression.

1)2)3)

4)5)6)

7)8)9)

HW: p.514#10-42 even

Lesson 15 (PH Text 8.8): Factoring by Grouping

Objectives: to factor higher-degree polynomialsby 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:

27x3 – 3xy2
4m3 – 48m2 + 144m
18x2 – 12x + 2
8x2y3 + 4x2y2 – 12x2y
2d5 – 162d

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

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 – 9d8.25r – r3 – r2s + 25s

9.49n2 – 9m2 + 24m – 16

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; 9b221.4c2 + 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 – x2.2x + 1 and 2x – 1

3. 3t – 4 and 4 – 3t 4.5y – 2 and 5y + 2

HW: p.519 #10-28 even, 35

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.

1.2.3.

4.5.

6.7.

HW: p.655#8-32 even, 43