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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.

 

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