Unit 6: Polynomials6. 1 Objectives

The student will be able to:

1. multiply monomials.

2.simplify expressions with monomials.

Hernandez – Henry Ford High School

A monomial is a1. number,

2. variable, or

3. a product of one or more numbers and variables.

Examples:

5

y

3x2y3

Why are the following not monomials?

x + yaddition

division

2 - 3a

subtraction

x

y

Multiplying MonomialsWhen multiplying monomials, you

ADD the exponents.

1) x2 • x4

x2+4

x6

2) 2a2y3 • 3a3y4

6a5y7

Simplify m3(m4)(m)

1. m7

2. m8

3. m12

4. m13

Power of a PowerWhen you have an exponent with an

exponent, you multiply those exponents.

1) (x2)3

x2• 3

x6

2) (y3)4

y12

Simplify (p2)4

1. p2

2. p4

3. p8

4. p16

Power of a ProductWhen you have a power outside of the

parentheses, everything in the parentheses is raised to that power.

1) (2a)3

23a3

8a3

2) (3x)2

9x2

Simplify (4r)3

1. 12r3

2. 12r4

3. 64r3

4. 64r4

Power of a MonomialThis is a combination of all of the other

rules.

1) (x3y2)4

x3• 4 y2• 4

x12 y8

2) (4x4y3)3

64x12y9

Simplify (3a2b3)4

1. 12a8b12

2. 81a6b7

3. 81a16b81

4. 81a8b12

6.2 ObjectivesThe student will be able to:

1. divide monomials.

2.simplify negative exponents.

When dividing monomials, subtract the exponents.

1.

2.

Dividing Monomials

= m6 n3

5

2

b

b

7 5

2

m n

mn

= b5-2b b b b b

b b

= b3

= m7-1n5-2

= xy

= 9a3b2

8 38-5 3-1

5

81a b9a b

9a b

672362

73 yx

yx

yx

Simplify 2 7

5

64g h

16gh

1. 48g2h2

2. 48gh2

3. 4g2h2

4. 4gh2

= 1m0n

Here’s a tricky one!

What happened to the m?

= n

3 3

3 2

3m n 3m m m n n n

3m n 3m m m n n

They canceled out!

There are no m’s left over!This leads us to our next rule…

3

31

m

m

Zero ExponentsAnything to the 0 power is equal to 1.

a0 = 1True or False?

Anything divided by itself equals one.True!

See for yourself!

3

3

3 x m1 1 1

3 x m

0 0 03 1 x 1 m 1

A negative exponent means you move the base to the other side of the fraction and make the exponent positive.

-n n-n n

n -n

a 1 1 aa or = a

1 a a 1

Negative Exponents

Notice that the base with the negative exponent moved and became positive!

Simplify.

-4 04

1x and y 1

x

6. x-4 y0

You can not have negative or zero exponents in your answer.

4 4

1 1

x

1

x

1. p2

2. p12

3. .

4. .

Simplify -7

-5

p

p

12

1

p

2

1

p

Simplify.4 3 2 41

r s6

21rs

6

2

r

6s

You can’t leave the negative exponent!

There is another way of doing this without negative exponents.

If you don’t want to see it, skip the next slide!!!

Simplify (alternate version).Look and see (visualize) where you have the larger exponent and leave the variable in that location. Subtract the smaller exponent from the larger one.

In this problem, r is larger in the numerator and s is larger in the denominator.

4 2 4-3

3 4 4-2

3r s 1r

18r s 6s 2

r

6s

Notice that you did not have to work with negative exponents! This method is quicker!

Simplify.

7

1

x

Get rid of the negative exponent.

3

4 3 4

x 1

x x x

8

s

r

Get rid of the negative exponents.

Simplify.

5 2 3

3 3 5 3 2

r s s

r s r r s

4

12

y

64x

Get rid of the parentheses.

Simplify.

23 2

32

x y

4x

Get rid of the

negative exponents.

4

3 6 6

y

4 x x

1. .

2. .

3. .

4. .

Simplify

22 1

33

-3x

2x

y

y

11

9x

8y

11

-9x

8y

7

-9x

8y

7

9x

8y

6.3 ObjectiveThe student will be able to:

express numbers in scientific and decimal notation.

How wide is our universe?210,000,000,000,000,000,000,000 miles

(22 zeros)

This number is written in decimal notation. When numbers get this large,

it is easier to write them in scientific notation.

Scientific Notation

A number is expressed in scientific notation when it is in the form

a x 10n

where a is between 1 and 10

and n is an integer

Write the width of the universe in scientific notation.

210,000,000,000,000,000,000,000 miles

Where is the decimal point now?

After the last zero.

Where would you put the decimal to make this number be between 1 and 10?

Between the 2 and the 1

2.10,000,000,000,000,000,000,000.

How many decimal places did you move the decimal?

23When the original number is more than 1,

the exponent is positive.The answer in scientific notation is

2.1 x 1023

1) Express 0.0000000902 in scientific notation.

Where would the decimal go to make the number be between 1 and 10?

9.02The decimal was moved how many places?

8When the original number is less than 1, the

exponent is negative.9.02 x 10-8

Write 28750.9 in scientific notation.

1. 2.87509 x 10-5

2. 2.87509 x 10-4

3. 2.87509 x 104

4. 2.87509 x 105

2) Express 1.8 x 10-4 in decimal notation.0.00018

3) Express 4.58 x 106 in decimal notation.

4,580,000

On the graphing calculator, scientific notation is done with the button.

4.58 x 106 is typed 4.58 6

4) Use a calculator to evaluate: 4.5 x 10-5

1.6 x 10-2

Type 4.5 -5 1.6 -2

You must include parentheses if you don’t use those buttons!!

(4.5 x 10 -5) (1.6 x 10 -2)

0.0028125Write in scientific notation.

2.8125 x 10-3

5) Use a calculator to evaluate: 7.2 x 10-9

1.2 x 102

On the calculator, the answer is:6.E -11

The answer in scientific notation is 6 x 10 -11

The answer in decimal notation is 0.00000000006

6) Use a calculator to evaluate (0.0042)(330,000).

On the calculator, the answer is1386.

The answer in decimal notation is

1386

The answer in scientific notation is

1.386 x 103

7) Use a calculator to evaluate (3,600,000,000)(23).

On the calculator, the answer is:

8.28 E +10

The answer in scientific notation is

8.28 x 10 10

The answer in decimal notation is

82,800,000,000

Write (2.8 x 103)(5.1 x 10-7) in scientific notation.

1. 14.28 x 10-4

2. 1.428 x 10-3

3. 14.28 x 1010

4. 1.428 x 1011

Write in PROPER scientific notation.(Notice the number is not between 1 and 10)

8) 234.6 x 109

2.346 x 1011

9) 0.0642 x 104

on calculator: 642

6.42 x 10 2

Write 531.42 x 105 in scientific notation.

1. .53142 x 102

2. 5.3142 x 103

3. 53.142 x 104

4. 531.42 x 105

5. 53.142 x 106

6. 5.3142 x 107

7. .53142 x 108

6. 4 ObjectivesThe student will be able to:

1. find the degree of a polynomial.

2. arrange the terms of a polynomial in ascending or descending order.

SOL: noneDesigned by Skip Tyler, Varina High School

What does each prefix mean?mono

one

bi

two

tri

three

What about poly?one or more

A polynomial is a monomial or a sum/difference of monomials.

Important Note!!An expression is not a polynomial if there is a variable in the denominator.

State whether each expression is a polynomial. If it is, identify it.

1) 7y - 3x + 4

trinomial

2) 10x3yz2

monomial

3)

not a polynomial2

57

2y

y

Which polynomial is represented by

X2

1

1

X

X

X

1. x2 + x + 1

2. x2 + x + 2

3. x2 + 2x + 2

4. x2 + 3x + 2

5. I ’ve got no idea!

The degree of a monomial is the sum of the exponents of the variables.

Find the degree of each monomial.1) 5x2

2

2) 4a4b3c

8

3) -3

0

To find the degree of a polynomial, find the largest degree of the terms.

1) 8x2 - 2x + 7

Degrees: 2 1 0

Which is biggest? 2 is the degree!

2) y7 + 6y4 + 3x4m4

Degrees: 7 4 8

8 is the degree!

Find the degree of x5 – x3y2 + 4

1. 0

2. 2

3. 3

4. 5

5. 10

A polynomial is normally put in ascending or descending order.

What is ascending order?

Going from small to big exponents.

What is descending order?

Going from big to small exponents.

Put in descending order:

1) 8x - 3x2 + x4 - 4

x4 - 3x2 + 8x - 4

2) Put in descending order in terms of x:

12x2y3 - 6x3y2 + 3y - 2x

-6x3y2 + 12x2y3 - 2x + 3y

3) Put in ascending order in terms of y: 12x2y3 - 6x3y2 + 3y - 2x

-2x + 3y - 6x3y2 + 12x2y3

4) Put in ascending order:5a3 - 3 + 2a - a2

-3 + 2a - a2 + 5a3

Write in ascending order in terms of y:x4 – x3y2 + 4xy – 2x2y3

1. x4 + 4xy – x3y2– 2x2y3

2. – 2x2y3 – x3y2 + 4xy + x4

3. x4 – x3y2– 2x2y3 + 4xy

4. 4xy – 2x2y3 – x3y2 + x4

6. 5 ObjectivesThe student will be able to:

1. add and subtract polynomials.

1. Add the following polynomials:(9y - 7x + 15a) + (-3y + 8x - 8a)

Group your like terms.

9y - 3y - 7x + 8x + 15a - 8a

6y + x + 7a

Combine your like terms.

3a2 + 3ab + 4ab - b2 + 6b2

3a2 + 7ab + 5b2

2. Add the following polynomials:(3a2 + 3ab - b2) + (4ab + 6b2)

Add the polynomials.

+X2

11XX

XYYY

YY

1 11

XYY

Y 111

1. x2 + 3x + 7y + xy + 8

2. x2 + 4y + 2x + 3

3. 3x + 7y + 8

4. x2 + 11xy + 8

Line up your like terms. 4x2 - 2xy + 3y2

+ -3x2 - xy + 2y2

_________________________

x2 - 3xy + 5y2

3. Add the following polynomials using column form:

(4x2 - 2xy + 3y2) + (-3x2 - xy + 2y2)

Rewrite subtraction as adding the opposite.

(9y - 7x + 15a) + (+ 3y - 8x + 8a)

Group the like terms.

9y + 3y - 7x - 8x + 15a + 8a

12y - 15x + 23a

4. Subtract the following polynomials:(9y - 7x + 15a) - (-3y + 8x - 8a)

Rewrite subtraction as adding the opposite.

(7a - 10b) + (- 3a - 4b)Group the like terms.

7a - 3a - 10b - 4b4a - 14b

5. Subtract the following polynomials:(7a - 10b) - (3a + 4b)

Line up your like terms and add the opposite.

4x2 - 2xy + 3y2

+ (+ 3x2 + xy - 2y2)--------------------------------------

7x2 - xy + y2

6. Subtract the following polynomials using column form:

(4x2 - 2xy + 3y2) - (-3x2 - xy + 2y2)

Find the sum or difference.(5a – 3b) + (2a + 6b)

1. 3a – 9b

2. 3a + 3b

3. 7a + 3b

4. 7a – 3b

Find the sum or difference.(5a – 3b) – (2a + 6b)

1. 3a – 9b

2. 3a + 3b

3. 7a + 3b

4. 7a – 9b

6.6 ObjectivesThe student will be able to:

1. multiply a monomial and a polynomial.

add the exponents!

1) Simplify: 5(7n - 2)

Use the distributive property.

5 • 7n

35n - 10

Review: When multiplying variables,

- 5 • 2

3(8 12)

4a a 2) Simplify:

6a2 + 9a

3) Simplify: 6rs(r2s - 3) 6rs • r2s

6r3s2 - 18rs

38

4a a

312

4a

- 6rs • 3

4) Simplify: 4t2(3t2 + 2t - 5)

12t4

5) Simplify: - 4m3(-3m - 6n + 4p)

12m4

+ 8t3 - 20t2

+ 24m3n - 16m3p

6) Simplify: (27x2 - 6x + 12)

16x3 - 28x2 + 4x

Fooled ya, didn’t I?!? Ha! Ha!

Here’s the real answer!

-9x3 + 2x2 - 4x

Simplify 4y(3y2 – 1)

1. 7y2 – 1

2. 12y2 – 1

3. 12y3 – 1

4. 12y3 – 4y

Simplify -3x2y3(y2 – x2 + 2xy)

1. -3x2y5 + 3x4y3 – 6x3y4

2. -3x2y6 + 3x4y3 – 6x2y3

3. -3x2y5 + 3x4y3 – 6x2y3

4. 3x2y5 – 3x4y3 + 6x3y4

6.7 ObjectiveThe student will be able to:

multiply two polynomials using the FOIL method, Box method and the

distributive property.

There are three techniques you can use for multiplying polynomials.

The best part about it is that they are all the same! Huh? Whaddaya mean?

It’s all about how you write it…Here they are!1)Distributive Property2)FOIL3)Box Method

Sit back, relax (but make sure to write this down), and I’ll show ya!

1) Multiply. (2x + 3)(5x + 8)

Using the distributive property, multiply 2x(5x + 8) + 3(5x + 8).

10x2 + 16x + 15x + 24

Combine like terms.

10x2 + 31x + 24

A shortcut of the distributive property is called the FOIL method.

The FOIL method is ONLY used when you multiply 2 binomials. It is an

acronym and tells you which terms to multiply.

2) Use the FOIL method to multiply the following binomials:

(y + 3)(y + 7).

(y + 3)(y + 7). F tells you to multiply the FIRST

terms of each binomial.

y2

(y + 3)(y + 7). O tells you to multiply the OUTER

terms of each binomial.

y2 + 7y

(y + 3)(y + 7). I tells you to multiply the INNER

terms of each binomial.

y2 + 7y + 3y

(y + 3)(y + 7). L tells you to multiply the LAST

terms of each binomial.y2 + 7y + 3y + 21

Combine like terms.

y2 + 10y + 21

Remember, FOIL reminds you to multiply the:

First terms

Outer terms

Inner terms

Last terms

The third method is the Box Method. This method works for every problem!

Here’s how you do it. Multiply (3x – 5)(5x + 2)

Draw a box. Write a polynomial on the top and side of a box. It does not matter which goes where.

This will be modeled in the next problem along with

FOIL.

3x -5

5x

+2

3) Multiply (3x - 5)(5x + 2)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

15x2 - 19x – 10

3x -5

5x

+2

15x2

+6x

-25x

-10

You have 3 techniques. Pick the one you like the best!

15x2

+6x-25x-10

4) Multiply (7p - 2)(3p - 4)

First terms:

Outer terms:

Inner terms:

Last terms:

Combine like terms.

21p2 – 34p + 8

7p -2

3p

-4

21p2

-28p

-6p

+8

21p2

-28p-6p+8

Multiply (y + 4)(y – 3)1. y2 + y – 12

2. y2 – y – 12

3. y2 + 7y – 12

4. y2 – 7y – 12

5. y2 + y + 12

6. y2 – y + 12

7. y2 + 7y + 12

8. y2 – 7y + 12

Multiply (2a – 3b)(2a + 4b)1. 4a2 + 14ab – 12b2

2. 4a2 – 14ab – 12b2

3. 4a2 + 8ab – 6ba – 12b2

4. 4a2 + 2ab – 12b2

5. 4a2 – 2ab – 12b2

5) Multiply (2x - 5)(x2 - 5x + 4)You cannot use FOIL because they are not BOTH binomials. You must use the

distributive property.

2x(x2 - 5x + 4) - 5(x2 - 5x + 4)

2x3 - 10x2 + 8x - 5x2 + 25x - 20

Group and combine like terms.

2x3 - 10x2 - 5x2 + 8x + 25x - 20

2x3 - 15x2 + 33x - 20

x2 -5x +4

2x

-5

5) Multiply (2x - 5)(x2 - 5x + 4) You cannot use FOIL because they are not BOTH

binomials. You must use the distributive property or box method.

2x3

-5x2

-10x2

+25x

+8x

-20

Almost done!Go to

the next slide!

x2 -5x +4

2x

-5

5) Multiply (2x - 5)(x2 - 5x + 4) Combine like terms!

2x3

-5x2

-10x2

+25x

+8x

-20

2x3 – 15x2 + 33x - 20

Multiply (2p + 1)(p2 – 3p + 4)1. 2p3 + 2p3 + p + 4

2. y2 – y – 12

3. y2 + 7y – 12

4. y2 – 7y – 12