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Polynomialstwo PDFs of polynomials math study guide. If possible can you do a step by step so I can study from it. It’s highschool level

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Section 4.1 Introduction to Polynomials and Polynomial Functions 4-1

Copyright © 2015 Pearson Education, Inc.

Polynomial Expressions

ESSENTIALS

A monomial is a constant or a constant times some variable or variables raised to powers

that are nonnegative integers.

A polynomial is a monomial or a combination of sums and/or differences of monomials.

Examples of polynomials in one variable: 3 2 4 , 2 1, 5 6 ay xx

Examples of polynomials in several variables: 72 2 3 2 ,2 3,5 6 4 x y x yx y z

The terms of a polynomial are separated by + signs.

Names for certain types of polynomials:

Type

Definition:

Polynomial of Examples

Monomial One term 2 48 2 53 3, 2 , 4 , 7 , 2 x y ab xyz

Binomial Two terms 22 2 2 3 3 5, , 6 7 , 4 x a b x x x y xy

Trinomial Three terms 2 2 22 2

x 6 9, 2 7 3 , 3 4 x a ab b x y xy xy

The coefficient is the part of a term that is a constant factor. A constant term is a term

that contains only a number and no variable.

The degree of a term is the sum of the exponents of the variables, if there are variables.

The degree of a polynomial is the same as the degree of its term of highest degree.

The leading term of a polynomial is the term of highest degree. Its coefficient is called

the leading coefficient.

Descending order: arrangement of terms so that exponents decrease from left to right.

Ascending order: arrangement of terms so that exponents increase from left to right.

Example

Identify the terms, the degree of each term, and the degree of the polynomial

3 2 6 4 53. x x x Then identify the leading term, the leading coefficient, and the

constant term. Finally, write the polynomial in both descending and ascending order.

Descending order:

3 2 4 3 65 x x x

Ascending order:

2 3 56 3 4 x x x

Term 6x 3 4x 5 2 3x

Degree of term 1 3 0 2

Degree of polynomial 3

Leading term 3 4x

Leading Coefficient 4

Constant term 5

4-2 Section 4.1 Introduction to Polynomials and Polynomial Functions

Copyright © 2015 Pearson Education, Inc.

GUIDED LEARNING

EXAMPLE 1 YOUR TURN 1

Identify the terms, the degree of each term, and the

degree of the polynomial

5 39 6 2 4 12 10. y yy y Then identify the

leading term, the leading coefficient, and the

constant term.

Term 5 6y 2y 3 4y

9 12y 10

Degree of

term 3 0

Degree of

polynomial

Leading

term

9 12y

Leading

Coefficient

Constant

term

Identify the terms, the degree of each

term, and the degree of the polynomial

473 12 7 6 8. yyy Then identify the

leading term, the leading coefficient,

and the constant term.

Term

Degree of

term

Degree of

polynomial

Leading

term

Leading

Coefficient

Constant

term

EXAMPLE 2 YOUR TURN 2

Consider 5 93 6 12 4 2 10. y yyy Arrange in

descending order and then in ascending order.

Descending order: 5 6 2 y y

Ascending order: 3 10 4 y

Consider 473 12 7 6 8. yyy

Arrange in descending order and then

in ascending order.

Descending order:

Ascending order:

YOUR NOTES Write your questions and additional notes.

Section 4.1 Introduction to Polynomials and Polynomial Functions 4-3

Copyright © 2015 Pearson Education, Inc.

Evaluating Polynomial Functions

ESSENTIALS

To evaluate a polynomial, substitute a number for the variable.

Example

For the polynomial function 4 Px x x 2 3 1, find P 4 .

4

4

2 31

4 24 34 1

2 256 12 1

512 12 1

501

Px x x

P

GUIDED LEARNING

EXAMPLE 1 YOUR TURN 1

For the polynomial function

2 fx x x 3 5 4, find f 1 .

2

2

3 54

1 31 51 4

31 5 1 4

5 4

fx x x

f

For the polynomial function

3 2 fx x x x 6 4 2 10, find f 2 .

EXAMPLE 2 YOUR TURN 2

When an object is launched upward with an

initial velocity of 25 m/sec and from the top

of a bridge that is 30 m high, its altitude, in

meters, after t seconds is given by the

polynomial function 2 A t tt 30 25 4.9 .

Find the altitude of the ball after 3 seconds.

We substitute 3 for t.

2

2

30 25 4.9

3 30 25 3 4.9 3

30 25 3 4.9 9

30 75

m

At t t

A

The polynomial function

3 2 Fx x x x 0.00016 0.0096 0.367 10.8

can be used to estimate the fuel economy, in

miles per gallon (mpg), for a particular

vehicle traveling x miles per hour (mph).

What is the gas mileage for this vehicle at

40 mph

YOUR NOTES Write your questions and additional notes.

4-4 Section 4.1 Introduction to Polynomials and Polynomial Functions

Copyright © 2015 Pearson Education, Inc.

Adding Polynomials

ESSENTIALS

Similar, or like, terms are terms that have the same variable(s) raised to the same

power(s).

Polynomials are added by combining, or collecting, like terms.

Example

Add: 3 32 4 5 3 6 2 7. xx xx

3 32 32 32 4 5 3 6 2 7 46 2 5 37 2 2 5 4 xx xx xxx xxx

GUIDED LEARNING

EXAMPLE 1 YOUR TURN 1

Collect like terms: 2 2 2 12 11 3 15 6. xx xx

2 2

2

2 12 11 3 15 6

2 3 12 15 11 6

3

xx xx

x x

x

Collect like terms:

2 2 4 3 1 8 10 6. xx x x

EXAMPLE 2 YOUR TURN 2

Add: 3 33 3 4 2 5 6 5. x xy y x xy y

3 33 3

3 3

3

42 565

45 26 15

9

x xy y x xy y

x xy y

x

Add:

23 2 4

23 2 4

12 3 2

5 10 .

xy xy xy

x y xy xy

EXAMPLE 3 YOUR TURN 3

Add using columns:

3 32 5 6 8 2 4 4. yy yy

In order to use columns to add, we write the

polynomials one under the other, listing like terms

under one another and leaving spaces for missing

terms.

3 2

3 2

3

5 68 4

24 4

3 6

6

y y

y y

y y

y

y

Add using columns:

43 4 5 2 5 7 3 10 . yy yy

YOUR NOTES Write your questions and additional notes.

Section 4.1 Introduction to Polynomials and Polynomial Functions 4-5

Copyright © 2015 Pearson Education, Inc.

Subtracting Polynomials

ESSENTIALS

If the sum of two polynomials is 0, the polynomials are opposites, or additive inverses.

The Opposite of a Polynomial

The opposite of a polynomial P can be written as P or, equivalently, by replacing

each term in P with its opposite.

To subtract a polynomial, we add its opposite.

Examples

Write two equivalent expressions for the opposite of 3 2 3 2 10 1. xx x

The opposite can be written with parentheses as 3 2 3 2 10 1 . xx x

The opposite can be written without parentheses by replacing each term by its

opposite: 3 2 3 2 10 1. xx x

Subtract: 2 2 4 3 1 10 6 8 . xx xx

22 2 2

2 2

2

4 3 1 10 6 8 4 3 1 10 6 8

4 3 1 10 6 8

14 9 7

xx xx xx xx

xx xx

x x

GUIDED LEARNING

EXAMPLE 1 YOUR TURN 1

Write two equivalent expressions for the

opposite of 2 3 2 4 12. x xy y

The opposite can be written with parentheses

as 2 3 2 4 12 . x xy y

The opposite can be written without

parentheses by replacing each term by its

opposite: 2 3 12. x

Write two equivalent expressions for the

opposite of 2 6 5 16 20. xy xy y

EXAMPLE 2 YOUR TURN 2

Subtract: 3 7 5 1 15 7 . x xy xy

We add the opposite of the polynomial being

subtracted.

3

3

3

7 5 1 15 7

7 5 1 15 7

7 5 1 15 7

10

x xy xy

x xy xy

x xy xy

xy

Subtract: 2 33 2 33 6 5 8 3 11 . x xy x xy

4-6 Section 4.1 Introduction to Polynomials and Polynomial Functions

Copyright © 2015 Pearson Education, Inc.

EXAMPLE 3 YOUR TURN 3

Subtract: 1 51 3 31 2 2 . 4 82 4 82

xx xx

2 2

2 2

2

2

1 51 3 31

4 82 4 82

1 51 3 3

4 82 4 8

1 51 31

4 82 82

2 8 1

4 8

xx xx

xx xx

xx x

x x

x

Subtract:

7 25 2 11 2 2 . 9 36 9 36

xx xx

YOUR NOTES

Write your questions and additional notes.

Section 4.1 Introduction to Polynomials and Polynomial Functions 4-7

Copyright © 2015 Pearson Education, Inc.

Practice Exercises

Readiness Check

Choose the word, number, or expression that best completes each statement.

1. The expression 2 4 2 x is a __________________ with a leading coefficient of

binomial / trinomial

_____. The degree of the term 2 is _____.

2 / 4 0 / 2

2. The degree of the polynomial 2 2 3 5 10 4 xy xyz xy is _____. The leading

2 / 4

coefficient is ______. The polynomial has _____ terms.

5 / 3 4 / 3

3. The polynomial 2 12 3 15 x x is written in ____________________ order.

ascending / descending

4. To add or subtract polynomials you must add or subtract ____________________.

like terms / exponents

Polynomial Expressions

Identify the terms, the degree of each term, and the degree of the polynomial. Then identify

the leading term, the leading coefficient, and the constant term.

5. 3 4 6 83 x x x 6. 473 5 12 6 3 2 x xxx 7. 2 36 27 xy xyz x

Arrange in descending powers of x.

8. 4 2 15 3 2 x x x 9. 2 63 4 3 12 8 10 xx x x

Arrange in ascending powers of y.

10. 24 3 2 39 yy y 11. 2 432 3 5 26 x y xy x y

Evaluating Polynomial Functions

Find f 5 and f 1 for each polynomial function.

12. 3 2 fx x x 4 12 20 13. 2 3 f x xx 43 5

4-8 Section 4.1 Introduction to Polynomials and Polynomial Functions

Copyright © 2015 Pearson Education, Inc.

14. A firm determines that, when it sells x tablet computers, its total revenue, in dollars, is

2 Rx x x 280 0.4 . What is the total revenue from the sale of 62 tablet computers

Adding Polynomials

Collect like terms to write an equivalent expression.

15. 3 9 34 2 4 7 x xx x 16. 22 3 22 3 4 5 9 15 x y x xy x

Add.

17. 2 835 2 xyz x y xy xyz xy 18. 2 22 3 2 5 84 7 xx x x

Subtracting Polynomials

Write two expressions, one with parentheses and one without, for the opposite of each

polynomial.

19. 43 2 9325 xxx 20. 2 2 12 7 6 xy x y xyz

Subtract.

21. 2 2 4 52 2 49 xx xx 22. 3 2 32 7 4 5 3 17 a a aa

23. 7 2 22 7 6 2 32 x yz x y z x x x yz 24. 5 5 2 12 7 15 13 x xy x

Section 4.2 Multiplication of Polynomials 4-9

Copyright © 2015 Pearson Education, Inc.

Multiplication of Any Two Polynomials

ESSENTIALS

Multiplying Monomials:

To multiply two monomials, multiply the coefficients and multiply the variables using

the rules for exponents and the commutative and associative laws.

Multiplying Monomials and Binomials:

The distributive law is used to multiply polynomials other than monomials.

Multiplying Binomials:

To multiply binomials use the distributive law twice, first consider one of the binomials

as a single expression and multiply it by each term of the other binomial.

Multiplying Any Two Polynomials:

To find the product of two polynomials P and Q, multiply each term of P by every term

of Q and then collect like terms.

Examples

Multiply: 34 22 5 3. x y xy

34 22 3 2 4 2

32 42

5 6

5 3 53

15

15

x y xy x x y y

x y

x y

Multiply: 4 5 2. x x

2

4 5 2 4 5 4 2 Using the distributive law

20 8 Multiplying monomials

xx xx x

x x

Multiply: 2 2

x x 1 2 3.

2 2 2 22 2

22 2

22 2 2

422

4 2

1 2 3 1 2 1 3 Distributing the 1

2 1 3 1 Using a commutative law

2 2 1 3 3 1 Using the distributive law twice

2 2 3 3 Multiplying monomials

2 3 Collecting like terms

x x x xx x

xx x

xx x x

xxx

x x

Multiply: 3 2

xx x 4 5 3.

32 32 32

3 2 32

43 3 2

43 2

4 5 3 4 5 4 53

4 5 3 4 3 53

4 5 3 12 15

12 5 15

x x x x x xx x

xx xx x x x

x x xx x

xx x x

4-10 Section 4.2 Multiplication of Polynomials

Copyright © 2015 Pearson Education, Inc.

GUIDED LEARNING

EXAMPLE 1 YOUR TURN 1

Multiply and simpify: 23 5 7 3. a bc abc

23 5 2 35

11 3 5

7 3 73

21

a bc abc a a b b c c

a bc

Multiply and simplify:

35 4 12 4 . a b c a bc

EXAMPLE 2 YOUR TURN 2

Multiply: 23 4 24 5. aa a

23 4 23 4

3

24 5 24 5

8

aa a aa a

a

Multiply: 3 9 4 8. a a

EXAMPLE 3 YOUR TURN 3

Multiply: 2 3 x x 5 2.

2 3 2 32

32 2

32 3

5 2 5 52

52 5

2 25

x x x xx

xx x

xx x

Multiply: 4 5 x x 1 3 6.

EXAMPLE 4 YOUR TURN 4

Multiply: 3 2 3 2 2 2 4 5. xx xx

5 4

3

2

3

4

5 32

2

5 3

4

12 12

15 1 10

1

3 22

2 45

15 10 10

12 8

2 1 1 10 10 4 4

xx x

x

x x

x x

x x

x

x x

x x

x xx

Multiply: 3 2 2 4 6 5 1. x x xx

YOUR NOTES Write your questions and additional notes.

Section 4.2 Multiplication of Polynomials 4-11

Copyright © 2015 Pearson Education, Inc.

Product of Two Binomials Using the FOIL Method

ESSENTIALS

The FOIL Method

To multiply two binomials A B and C D :

1. Multiply First terms: AC

2. Multiply Outer terms: AD

3. Multiply Inner terms: BC

4. Multiply Last terms: BD

FOIL

A B C D AC AD BC BD

Example

Multiply: x x 6 3.

2

2

6 3 3 6 18 FOIL

3 18 Collecting like terms

x x x xx

x x

GUIDED LEARNING

EXAMPLE 1 YOUR TURN 1

Multiply: 2 1 4 5. x x

2 2145 8 45 x xx x

Multiply: 3 6 6 2. x x

EXAMPLE 2 YOUR TURN 2

Multiply: mm m 1 4 7.

2

2

2 2

2

147

4 47

4 7

3 4 3 47

3 4 21 28

mm m

m mm m

m m

m m mm m

mm m

Multiply: mmm 8 5 4.

YOUR NOTES Write your questions and additional notes.

4-12 Section 4.2 Multiplication of Polynomials

Copyright © 2015 Pearson Education, Inc.

Squares of Binomials

ESSENTIALS

Squaring a Binomial

2 2 2

2 2 2

2

2

A B A AB B

A B A AB B

Trinomials of the form 2 2 A AB B 2 or 2 2 A AB B 2 are called trinomial squares.

Example

Multiply: 2

x 4 .

2 2 2

2 2 2

2

2

4 2 44

8 16

A B A AB B

x xx

x x

GUIDED LEARNING

EXAMPLE 1 YOUR TURN 1

Multiply: 2

4 6. y

2 2 2 4 6 4 24 6 6 y yy

Multiply: 2

3 5. y

EXAMPLE 2 YOUR TURN 2

Multiply:

2

1 2 .

4

x

2 2 1 11 2 2 2 22

4 44

x xx

Multiply:

2

1 5 .

2

x

YOUR NOTES Write your questions and additional notes.

Section 4.2 Multiplication of Polynomials 4-13

Copyright © 2015 Pearson Education, Inc.

Products of Sums and Differences

ESSENTIALS

The Product of a Sum and a Difference

2 2 A BAB A B

The product of the sum and the difference of the same two terms is called a difference of

squares.

Example

Multiply: x x 3 3.

2 2

2 2

2

33 3

9

A BAB A B

xx x

x

GUIDED LEARNING

EXAMPLE 1 YOUR TURN 1

Multiply: 2 4 2 4. xy x xy x

2 2 2 42 4 2 4 xy x xy x xy x

Multiply: 2 2 5 1 5 1. xy xy

EXAMPLE 2 YOUR TURN 2

Multiply: 0.3 2.1 0.3 2.1 . mnmn

2 2 0.3 2.1 0.3 2.1 0.3 2.1 mnmn m n

Multiply: 0.6 1.2 0.6 1.2 . x yxy

4-14 Section 4.2 Multiplication of Polynomials

Copyright © 2015 Pearson Education, Inc.

YOUR NOTES Write your questions and additional notes.

Section 4.2 Multiplication of Polynomials 4-15

Copyright © 2015 Pearson Education, Inc.

Using Function Notation

ESSENTIALS

Example

Given 2 fx x x 6 7, find (a) f t 10 and (b) f t 10 .

a) 2 fx x x 6 7

2

2

10 6 7 10 Evaluating

6 17 Simplifying

f t t t ft

t t

b) 2 fx x x 6 7

2

2

2

10 10 6 10 7 Substituting 10 for

20 100 6 60 7 Multiplying

14 33 Simplifying

f tt t t x

tt t

t t

GUIDED LEARNING

EXAMPLE 1 YOUR TURN 1

Given 2 fx x 2 3, find (a) f a 1;

(b) f a 1 .

a)

2

2

2

2 3

1 2 31

2 4

fx x

fa a

a

b)

3

2

2

2 3

12 1 3

2 3

24 3

fx x

fa a

a a

Given 2 f xx x 5 , find (a) f a 4;

(b) f a 4 .

4-16 Section 4.2 Multiplication of Polynomials

Copyright © 2015 Pearson Education, Inc.

EXAMPLE 2 YOUR TURN 2

Given 2 3

g x xy 3 2, find gx gx .

23 23

2 2 3

23 2

3 23 2

3 2

23 22

g x g x xy xy

x y

x y

Given 2 6

g x xy 4 3, find gx gx .

YOUR NOTES Write your questions and additional notes.

Section 4.2 Multiplication of Polynomials 4-17

Copyright © 2015 Pearson Education, Inc.

Practice Exercises

Readiness Check

Classify each of the following statements as either true or false.

1. FOIL can be used whenever two binomials are multiplied.

2. Given fx x 2 1, fy fy 8 8.

3. A trinomial square is of the form 2 2 A B .

4. When multiplying polynomials, the distributive law can always be used.

Multiplying Monomials

Multiply.

5. 5 3 2 4 x x 6. 67 22 5 3 x y xy 7. 4 2 9 3 abc a bc

Multiplying Monomials and Binomials

Multiply.

8. 64 2 x 9. 2 43 7 aa a 10. 23 4 57 6 xy xy xy

Multiplying Any Two Polynomials

Multiply.

11. 2 35 1 x x 12. 2

x xx 5 2 13. 2

x xx 43 5

The Product of Two Binomials: FOIL

Multiply.

14. 1 1

6 2

x x

15. 4 23 1 x x 16. 1.3 3 2.4 5 x x

4-18 Section 4.2 Multiplication of Polynomials

Copyright © 2015 Pearson Education, Inc.

Squares of Binomials

Multiply.

17. 2

x 6 18. 2

2 3 x y 19.

2 3

x y 2

Products of Sums and Differences

Multiply.

20. a a 10 10 21. 36 36 x x 22. 2 2 4 4 ab ab

Function Notation

23. Let Px x 2 5 and

2 Qx x x 4 3 1.

Find Px Qx .

24. Given 2 fx x x 3 9, find

f t 4 .

25. Given 2 fx x 2 3, find

f a h .

26. Given 2 f x xx 64 , find

f a 7.