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Chapter 1.4
POLYNOMIAL
Expressions
1
represented only by letters
and whose values may be
arbitrarily chosen depending
on the situation.
Types of Quantities
Variables
2
a quantity whose value is fixed
and may not be changed during
a particular discussion.
Types of Quantities
Constant
3
Example 1.4.1
Consider the formula
- force
- mass
- acceleration
, , and are variables.
F ma
F
m
a
F m a4
Example 1.4.1
In the formula for computing the
circumference of a circle
and are constants,
while is a varia
2
2
ble.
r
r
5
Algebraic Expressions
Any combination of numbers and
symbols related by the operations
from the previous sections will be
algebraic excalle pressd an ion.
6
2 2
3 10
2 3
2
are algebraic expressions
x y
xx y xy x
y
x y
x y
Example 1.4.2
7
Algebraic Expressions
2
Each part of the expression separated by + or ,
together with its sign, is called a of the
algebraic expression
te
.
2 3 2 5
rm
x xy xy
8
Types of Algebraic
Expressions
An algebraic expression consisting of
1 term - monomial
2 terms - binomial
3 terms - trinomial
any number of terms - multinomial
9
Algebraic Expressions
Each factor of a term may be call
coefficient
ed
the of the others.
4
4
4
4
4
In 3 ,
3 is the coefficient of .
3 is the coefficient of .
3 is the coef
numerical coefficien
ficient of .
3 is the .
is the
t
literal coefficient.
u v
u v
u v
v u
u v 10
Algebraic Expressions
2 2
Terms having the same literal
coefficient similar
terms
s are called
.
3 2 5x xy x y
11
Polynomial Expressions
A is an algebraic
expression involving only non-
negative integra
polynomi
l powers
al in
of .
t
t
12
Example 1.4.3
2 3Is 2 3 1 a polynomial in ?
Is it a polynomial in ?
Is it a polynomial in ?
xt yt t
x
y
13
Polynomials
2
2 2
A polynomial in can be expressed
as .
2 1
A polynomial in and can be
expressed , as .
,
P t
Q
t
P t t t
x y
Q x y x
x
y
y
14
Polynomials
The notations can be used to
find the value of the expression
when the value of the variable
is given.
15
Example 1.4.4
1If , then
2
14 4 2
2
10 0 0
2
S t gt
S g g
S g
16
Example 1.4.4
If 7, then
1 7
7
is constant polynomial a .
C x
C
C
C
17
Polynomials
The of a term of a polynomial
is the exponent of the
variable.
The of a
degree
in a variable
degree in
some variab
term of a polynomial
is the sum of the degrees
of all variables in that t
les
erm.
18
Example 1.4.5
2
3 4
4 10
The degree of in is 2.
The degree of in is 4.
The degree of in and is 14.
x y x
x y y
x y x y
19
Polynomials
The degree of the polynomial in some
variables is the de highestgree of its term of
degree in those variables.
20
Example 1.4.6
3
4 3 2 2 3
1. 3 2
The degree of in is 3.
2. , 3 2
The degree of in is 4.
The degree of in is 3.
The degree of in and is 4.
P x x x
P x
Q x y x x y x y y
Q x
Q y
Q x y
21
Addition of Polynomials
To add polynomials, w add
similar t
e
erms.
3 2
3 2
3 2
If 2 1 and
3 5 10
2 5 9
P x x x
Q x x x x
P x Q x x x x
22
Example 1.4.7
Add the following:
5
2 4 3
2
1. 2 3 7
3 5
P x x x x
Q x x x x
P x Q x 5x 4x 33x 2x 3x 2
3 2 2 3
3 2 2 3
2. , 2 3 4 7 5
, 6 7 10
P x y x x y xy y
Q x y y xy x y x23
Multiplication of Polynomials
distributive property
To multiply two polynomials, we
apply the ,
the , and laws of exponents add
similar terms.
24
Example 1.4.8
2 2 2
3 2
3 2
3 2
Perform the indicated operations.
1. 2 5 3 2 2 3 2 5 3 2
6 4 15 10
6 4 15 10
6 15 4 10
t t t t t
t t t
t t t
t t t
25
Example 1.4.8
2 2
3 4 2 2 2 2
3 4 2 2 2
2. 2 3 2 4
6 4 8 3 2 4
6 4 10 3 4
x y xy x y
x y x x y xy x y y
x y x x y xy y
26
Special Products
2 2
Binomial Sum and Difference
x y x y x y
27
Example 1.4.9
2
3 3 2 6
2 3 2 3 4 2 6
Perform the indicated operation.
Use special products.
1. 2 2 4
2. 3 3 9
3. 5 2 5 2 25 4
x x x
x y x y x y
a b c a b c a b c
2 2x y x y x y
28
Special Products
2 2 2
Square of a Binomial
2x y x xy y
29
Example 1.4.10
2 2
2 2 2
22 4 2 2 2
Perform the indicated operation.
Use special products.
1. 3 6 9
2. 2 3 4 12 9
3. 5 10 25
x x x
a b a ab b
p q r p q p qr r
2 2 22x y x xy y
30
Special Products
3 3 2 2 3
3 3 2 2 3
3
Cube
3
3 3
of a Binomial
x y x x y y y
x y x x y xy y
31
Example 1.4.11
3 3
3 3 2
3
3 2 2 3
2 2 3
3 3 2 2 3
1. 2 6 12 8
2. 2 3
8
3
36 5
3
3 3
27
x x x x
a b
a
x y x x
a b ab
y y y
x y x x y xy
b
y
32
Special Products
2 2 3 3
2 2 3 3
Binomial and Trinomial
x y x xy y x y
x y x xy y x y
33
Example 1.4.12
2 2
2 3
2
3 3
2 2 3 3
3
1. 3 3 9 27
2. 3 2 9 6 4 27 8
x
x y x xy y x y
x y
x x x
y y
x xy y x y
y y
34
Factoring
pr
A p
odu
olynom
ct
of t
facto
wo or
ial is
more
if it can be
polynomials w
ra
it
ble
h
int
e
expressed
ger coeffi
as
cie
a
in
nts.
35
Factoring
A factorization is if each
factor
complete
prime facis a tor.
36
Special Products
2 2
2 2 2
2 2 3 3
2
x y x y x y
x y x xy y
x y x xy y x y
37
Example 1.4.13
4 4
2 24 4 2 2
2 2 2 2
2 2
Factor completely.x y
x y x y
x y x y
x y x y x y
38
Example 1.4.13
2 2
4 4 2 2
But these violate the restrictions
hence,
is a complete factorization.
x y x yi x yi
x y x y x y
x y x y x y x y
39
Example 1.4.14
2 2 3 3 4 5 2 2 3
3 4 6 2 4 6
2 22 3
2 3 2 3
Factor the following polynomials
completely.
1. 2 1 2
2. 25 25
5
5 5
x y x y x y x y xy xy
x y xd x x y d
x xy d
x xy d xy d
40
Example 1.4.14
22
22
2 2 2 2
2
3. 8 16 4
4. 4 12 9 2 3
5. 5 30 45 5 6 9
5 3
x x x
y y y
y yz z y yz z
y z
2 2 22x y x xy y
41
Example 1.4.14
3 3 3
2
33 3 3
2 2
6. 27 3
3 3 9
7. 64 4
4 4 16
b b
b b b
x y x y
x y x xy y
3 3 2 2x y x y x xy y
3 3 2 2x y x y x xy y
42
Example 1.4.14
2
2
2 2
8. 7 12 4 3
9. 42
7 6
10. 4 2 4
w w w w
x y x y
x y x y
x x y
43