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Section 5.1
Polynomials Addition
And Subtraction
OBJECTIVES
A
Classify polynomials.
OBJECTIVES
B
Find the degree of a
polynomial and write
descending order.
OBJECTIVES
C
Evaluate a polynomial.
OBJECTIVES
D
Add or subtract
polynomials.
OBJECTIVES
E
Solve applications
involving sums or
differences of
polynomials.
DEFINITION
Degree of a Polynomial in
One Variable
The degree of a polynomial in
one variable is the greatest
exponent of that variable.
DEFINITION
Degree of a Polynomial in
Several Variables
The greatest sum of the
exponents of the variables in
any one term of the polynomial.
RULES
Properties for Adding
Polynomials
If P, Q, and R are polynomials,
P +Q =Q + P
Commutative Property
RULES
Properties for Adding
Polynomials
If P, Q, and R are polynomials,
P + Q + R  = P + Q  + R
Associative Property
RULES
Properties for Adding
Polynomials
If P, Q, and R are polynomials,
P Q + R  = PQ + PR
Q + R P = QP + RP
RULES
Subtracting Polynomials
a – (b + c) = a – b – c
Chapter 5
Section 5.1A,B
Exercise #1
Classify as a monomial, binomial, or trinomial and
give the degree.
xy 3 z 4 – x 7
Binomial.
Degree is determined by comparing
Degree 1st term: x 1 y 3 z 4
1 + 3 + 4=8
Degree 2nd term: x 7
Degree 8
7
Chapter 5
Section 5.1D
Exercise #5
Add 6x 3 + 8x 2 – 6x – 4 and 6 – 3x + x 2 – 3x 3 .
METHOD 1
6x 3 + 8x 2 – 6x – 4
– 3x 3 + x 2 – 3x + 6
3x 3 + 9x 2 – 9x + 2
Add 6x 3 + 8x 2 – 6x – 4 and 6 – 3x + x 2 – 3x 3 .
METHOD 2

 
6 x 3 + 8x 2 – 6 x – 4 + 6 – 3 x + x 2 – 3 x 3

= 6x 3 + 8x 2 – 6x – 4 + 6 – 3x + x 2 – 3x 3
= 6x 3 – 3x 3 + 8x 2 + x 2 – 6x – 3x – 4 + 6
= 3x 3 + 9x 2 – 9x + 2
Chapter 5
Section 5.1D
Exercise #6
Subtract 8x 3 – 6x 2 + 5x – 3 from 5x 3 + 3x 2 + 3.
METHOD 1
–
5x 3 + 3x 2
+3
8x 3 – 6x 2 + 5x – 3
Subtract 8x 3 – 6x 2 + 5x – 3 from 5x 3 + 3x 2 + 3.
METHOD 1
–
5x 3 + 3x 2
+3
8x 3 – 6x 2 + 5x – 3
5x 3 + 3x 2
+
+3
Subtract 8x 3 – 6x 2 + 5x – 3 from 5x 3 + 3x 2 + 3.
METHOD 1
–
5x 3 + 3x 2
+3
8x 3 – 6x 2 + 5x – 3
+
5x 3 + 3x 2
+3
– 8x 3 + 6x 2 – 5x + 3
– 3x 3 + 9x 2 – 5x + 6
Subtract 8x 3 – 6x 2 + 5x – 3 from 5x 3 + 3x 2 + 3.
METHOD 2

 
5 x 3 + 3x 2 + 3 – 8 x 3 – 6 x 2 + 5 x – 3
= 5x 3 + 3x 2 + 3 – 8x 3 + 6x 2 – 5x + 3
= 5x 3 – 8x 3 + 3x 2 + 6x 2 – 5x + 3 + 3
= – 3x 3 + 9x 2 – 5x + 6

Section 5.2
Multiplication of
Polynomials
OBJECTIVES
A
Multiply a monomial by
a polynomial.
OBJECTIVES
B
Multiply two
polynomials.
OBJECTIVES
C
Use the FOIL method to
multiply two binomials.
OBJECTIVES
D
Square a binomial sum
or difference.
OBJECTIVES
E
Find the product of the
sum and the difference
of two terms.
OBJECTIVES
F
Use the ideas discussed
to solve applications.
RULES
Multiplication of Polynomials
If P, Q, and R are polynomials,
P
P Q =Q P
Q R  =  P Q  R
USING FOIL
To Multiply Two Binomials
(x + a)(x + b)
First Outside Inside Last
2
= x + bx + ax + ab
2
= x + (b + a)x + ab
RULE
To Square a Binomial Sum
(x + a)
2
= (x + a)(x + a)
2
= x + 2ax + a
2
RULE
To Square a Binomial Difference
2
(x – a)
= (x – a)(x – a)
= x
2
– 2ax + a
2
PROCEDURE
Sum and Difference of Same
Two Monomials
2
2
2
2
(x – a)(x + a) = x – a
(x + a)(x – a) = x – a
Chapter 5
Section 5.2B,C
Exercise #8a
Multiply.
 x – 2   x 2 – 4x – 5 
METHOD 1
x 2 – 4x – 5
x –2

– 2x 2 + 8x + 10
x 3 – 4x 2 – 5x
x 3 – 6x 2 + 3x + 10
Multiply.
 x – 2   x 2 – 4x – 5 
METHOD 2



= x x 2 – 4x – 5 +  – 2  x 2 – 4x – 5
= x 3 – 4x 2 – 5x – 2x 2 + 8x + 10
= x 3 – 4x 2 – 2x 2 – 5x + 8x + 10
= x 3 – 6x 2 + 3x + 10

Chapter 5
Section 5.2D
Exercise #9b
Multiply.
 3x – 4y 2
 x – a
2
=  3x  2 – 2(3x)(4y) +  4y  2
= 9x 2 – 24xy + 16y 2
= x 2 – 2ax + a 2
Chapter 5
Section 5.2E
Exercise #10
Multiply.
 3x + 4y  3x – 4y 
Product of Sum and Difference of
Same Two Monomials
 x + a  x – a  = x 2 – a 2
2
=  3x  –  4y 
= 9x 2 – 16y 2
2
Section 5.3
The Greatest Common
Factor and Factoring
by Grouping
OBJECTIVES
A
Factor out the greatest
common factor in a
polynomial.
OBJECTIVES
B
Factor a polynomial with
four terms by grouping.
GREATEST COMMON FACTOR
n
ax is the Greatest Common
monomial Factor (GCF) of a
polynomial in x if:
1. a is the greatest integer that
divides each coefficient.
GREATEST COMMON FACTOR
n
ax is the Greatest Common
monomial Factor (GCF) of a
polynomial in x if:
2. n is the smallest exponent of
x in all the terms.
PROCEDURE
Factoring by Grouping
1. Group terms with common
factors using the
associative property.
PROCEDURE
Factoring by Grouping
2. Factor each resulting
binomial.
PROCEDURE
Factoring by Grouping
3. Factor out the binomial
using the GCF, by the
distributive property.
Chapter 5
Section 5.3B
Exercise #12
Factor completely.
6x 7 + 6x 5 + 15x 4 + 15x 2

= 3x 2 2x 5 + 2x 3 + 5x 2 + 5

 


= 3x 2  2x 3 x 2 + 1 + 5 x 2 + 1 





= 3x 2  x 2 + 1 2x 3 + 5 




= 3x 2 x 2 + 1 2x 3 + 5

Section 5.4
Factoring Trinomials
OBJECTIVES
A
Factor a trinomial of the
2
form x + bx + c.
OBJECTIVES
B
Factor a trinomial of the
2
form ax + bx + c using
trial and error.
OBJECTIVES
C
Factor a trinomial of the
2
form ax + bx + c using
the ac test.
PROCEDURE
Factoring Trinomials
2
x + (b + a)x + ab
= (x + a)(x + b)
RULE
The ac Test
2
ax + bx + c is factorable only
if there are two integers
whose product is ac and
sum is b.
Chapter 5
Section 5.4A,B,C
Exercise #13b
Factor completely.
2x 2 + xy – 10y 2
The ac Method
Find
factors of ac (–20) whose sum is
(1) and replace the middle term
(xy).
= 2x 2 – 4xy + 5xy – 10y 2
= (2x 2 – 4xy) + (5xy – 10y 2)
= 2x(x – 2y) + 5y(x – 2y)
=  2x + 5y  x – 2y 
Section 5.5
Special Factoring
OBJECTIVES
A
Factor a perfect square
trinomial.
OBJECTIVES
B
Factor the difference of
two squares.
OBJECTIVES
C
Factor the sum or
difference of two cubes.
PROCEDURE
Factoring Perfect Square
Trinomials
2
2
2
x + 2 ax + a = (x + a)
2
2
2
x – 2 ax + a = (x – a)
PROCEDURE
Factoring the Difference of
Two Squares
2
2
x – a = (x + a)(x – a)
PROCEDURE
Factoring the Sum and
Difference of Two Cubes
3
3
2
2
x + a = (x + a)(x – ax + a )
3
3
2
2
x – a = (x – a)(x + ax + a )
Chapter 5
Section 5.5A
Exercise #15a
Factor completely.
16x 2 – 24xy + 9y 2
2
2
2
x – 2ax + a =  x – a 
Perfect Square Trinomial
= 16x 2 – 2  12xy  + 9y 2
=  4x – 3y  2
Chapter 5
Section 5.5
Exercise #16
Factor completely.
   
2
2
4
4
2
2
x – 16y = x
– 4y
Difference of Two Squares
x 2 – a 2 =  x + a  x – a 
=

x 2 + 4y 2

x 2 – 4y 2

Factor x 2 – 4y 2
=


x 2 + 4y 2  x + 2y  x – 2y 
Chapter 5
Section 5.5B
Exercise #17
Factor completely.
x 2 – 10x + 25 – y 2
2
2
 
x – 2ax + a = x – a
2
Perfect Square Trinomial
2
=  x – 5 – y 2
Difference of Two Squares
x 2 – a 2 =  x + a  x – a 
=   x – 5  + y   x – 5  – y 
=  x + y – 5  x – y – 5 
Chapter 5
Section 5.5c
Exercise #18a
Factor completely.
27x 3 + 8y 3
=  3x 3 +  2y 3
Sum of Two Cubes

x 3 + a 3 =  x + a  x 2 – ax + a 2

=  3x + 2y  9x 2 – 6xy + 4y 2


Section 5.6
General Methods of
Factoring
OBJECTIVES
A
Factor a polynomial
using the procedure
given in the text.
PROCEDURE
A General Factoring Strategy
1. Factor out the GCF, if
there is one.
2. Look at the number of terms
in the given polynomial.
PROCEDURE
A General Factoring Strategy
If there are two terms, look for:
Difference of Two Squares
2
2
x – a = (x + a)(x – a)
PROCEDURE
A General Factoring Strategy
If there are two terms, look for:
Difference of Two Cubes
3
3
2
2
x – a = (x – a)(x + ax + a )
PROCEDURE
A General Factoring Strategy
If there are two terms, look for:
Sum of Two Cubes
3
3
2
2
x + a = (x + a)(x – ax + a )
PROCEDURE
A General Factoring Strategy
If there are two terms, look for:
The sum of two squares,
2
2
x + a is not factorable.
PROCEDURE
A General Factoring Strategy
If there are three terms, look for:
Perfect square trinomial
2
2
2
x + 2ab + b = (x + a)
2
2
2
x – 2ab + b = (x – a)
PROCEDURE
A General Factoring Strategy
If there are three terms, look for:
Trinomials of the form
2
ax + bx + c (a > 0)
PROCEDURE
A General Factoring Strategy
Use the ac method or
trial and error.
If a < 0, factor out – 1 first.
PROCEDURE
A General Factoring Strategy
If there are four terms:
Factor by grouping.
PROCEDURE
A General Factoring Strategy
3. Check the result by
multiplying the factors.
Chapter 5
Section 5.6A
Exercise #20b
Factor completely.
48x 2y 2 – 72xy 3 + 27y 4

= 3y 2 16x 2 – 24xy + 9y 2

Perfect Square Trinomial
x 2 – 2ax + a 2 =  x – a 
2
= 3y 2  4x  2 – 2  12xy  +  3y  2 


2
2
= 3y  4x – 3y 
Chapter 5
Section 5.6A
Exercise #21
The ac Method
Find factors of ac (–12) whose sum is (–11) and replace the
middle term (–11xy).
Factor completely: 9x 3y – 33x 2y 2 – 12xy 3
= 3xy [3x 2 – 11xy – 4y 2]
= 3xy [3x 2 – 12xy + xy – 4y 2]
= 3xy [  3x 2 – 12xy  +  xy – 4y 2  ]
= 3xy [ 3x  x – 4y  + y  x – 4y  ]
= 3xy [  x – 4y  3x + y  ]
The ac Method
Find factors of ac (–12) whose sum is (–11) and replace the
middle term (–11xy).
Factor completely: 9x 3y – 33x 2y 2 – 12xy 3
= 3xy [3x 2 – 11xy – 4y 2]
= 3xy [3x 2 – 12xy + xy – 4y 2]
= 3xy [  3x 2 – 12xy  +  xy – 4y 2  ]
= 3xy [ 3x  x – 4y  + y  x – 4y  ]
= 3xy  3x + y  x – 4y 
Chapter 5
Section 5.6A
Exercise #22
Factor completely.
16x 3 – 12x 2 – 4x + 3
= 4x 2  4x – 3  – 1 4x – 3 


=  4x – 3  4x 2 – 1
Difference of Two Squares
x 2 – a 2 =  x + a  x – a 
=  4x – 3  2x + 1 2x – 1
Section 5.7
Solving Equations by
Factoring:
Applications
OBJECTIVES
A
Solve equations by
factoring.
OBJECTIVES
B
Use Pythagorean theorem
to find the length of one
side of a right triangle
when the lengths of the
other two sides are given.
OBJECTIVES
C
Solve applications
involving quadratic
equations.
PROCEDURE
1. Set equation equal to 0. O
2. Factor Completely.
F
3. Set each linear Factor
F
equal to 0 and solve each.
DEFINITION
Pythagorean Theorem
In symbols,
2
2
2
c =a +b
a
c
b
Chapter 5
Section 5.7A
Exercise #23b
Solve.
O
6x 2 + 7x = 3
6x 2 + 7x – 3 = 0
F  3x – 1 2x + 3  = 0
F 3x – 1 = 0
3x = 1
1
x=
3
or
or
2x + 3 = 0
2x = – 3
3
x=–
2
1
3
x= , –
3
2