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Polynomials
two 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
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
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
Constant term 5
4-2 Section 4.1 ? Introduction to Polynomials and Polynomial Functions
GUIDED LEARNING
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
constant term.
Term 5 ?6y ?2y 3 ?4y
9 12y 10
Degree of
term 3 0
Degree of
polynomial
term
9 12y
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
and the constant term.
Term
Degree of
term
Degree of
polynomial
term
Coefficient
Constant
term
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:
Section 4.1 ? Introduction to Polynomials and Polynomial Functions 4-3
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
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 .?
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?
4-4 Section 4.1 ? Introduction to Polynomials and Polynomial Functions
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
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 ? ?? ? ?
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
? ?? ? ?
? ? ??? ? ?
?
?
? ?
? ?
23 2 4
23 2 4
12 3 2
5 10 .
xy xy xy
x y xy xy
? ? ?
? ?
? ?? ? 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
? ?
?? ? ?
? ?
? ?
?
? ? ? ? 43 4 5 2 5 7 3 10 . yy yy ? ?? ? ?
Section 4.1 ? Introduction to Polynomials and Polynomial Functions 4-5
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
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
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
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
? ?? ? ? ?? ? ? ? ?? ? ? ? ?? ?
Section 4.1 ? Introduction to Polynomials and Polynomial Functions 4-7
Practice Exercises
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
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
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?
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
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
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
GUIDED LEARNING
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 ?
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 ?
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.
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 ? ? ? ??
Section 4.2 ? Multiplication of Polynomials 4-11
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
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
Multiply: ? ?? ? 2 1 4 5. x x ? ??
? ?? ? 2 2145 8 45 x ? ? ? ?? ? ? ? xx x
?
Multiply: ?3 6 6 2. x x ? ?? ? ?
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. ?? ? ? ?? ?
4-12 Section 4.2 ? Multiplication of Polynomials
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
Multiply: ? ?2
4 6. y ?
? ? ?? 2 2 2 4 6 4 24 6 6 y yy ? ? ?? ??
?
Multiply: ? ?2
? ? 3 5. y
Multiply:
2
1 2 .
4
x
? ? ? ? ? ? ?
2 2 1 11 2 2 2 22
4 44
x xx
? ?? ? ? ?? ? ? ? ?? ?? ? ?? ?
?
Multiply:
2
1 5 .
2
x
? ? ? ? ? ? ?
Section 4.2 ? Multiplication of Polynomials 4-13
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
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 ? ?
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
Section 4.2 ? Multiplication of Polynomials 4-15
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
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
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 ? ?? ? ?.
Section 4.2 ? Multiplication of Polynomials 4-17
Practice Exercises
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
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.