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5.3 – Polynomials and Polynomial Functions
Definitions
Term: a number or a product of a number and variables raised
to a power.
2
2
3, 5x ,  2 x, 9 x y
Coefficient: the numerical factor of each term.
5x 2 ,  2 x, 9 x 2 y
Constant: the term without a variable.
3,  6, 5, 32
Polynomial: a finite sum of terms of the form axn, where a is a
real number and n is a whole number.
15 x  2 x  5
3
2
21y 6  7 y 5  2 y 3  6 y
5.3 – Polynomials and Polynomial Functions
Definitions
Monomial: a polynomial with exactly one term.
2
ax ,
rt ,
4
2x ,
9m,
2
9x y
Binomial: a polynomial with exactly two terms.
x  8, r  3, 5 x 2  2 x, 2 x  9 x 2 y
Trinomial: a polynomial with exactly three terms.
x 2  x  8,
r 5  3r  3, 5 x 2  2 x  7
5.3 – Polynomials and Polynomial Functions
Definitions
The Degree of a Term with one variable is the exponent on
the variable.
5x 2  2,
2x 4  4,
9m  1
The Degree of a Term with more than one variable is
the sum of the exponents on the variables.
7x y  3,
2
2x y  6,
4
2
9mn z  10
5 4
The Degree of a Polynomial is the greatest degree of
the terms of the polynomial variables.
2 x  3 x  7  3,
3
2 x y  5x y  6 x  6
4
2
2
3
5.3 – Polynomials and Polynomial Functions
Practice Problems
Identify the degrees of each term and the degree of the
polynomial.
3a b  2ab  9b  4
3 0
6
6
6
5x  4 x  5x
2
1
3
3
2 4
2
3
5
4 x5 y 4  5x 4 y5  6 x3 y 3  2 xy
9
9
6
9
2
3
5.3 – Polynomials and Polynomial Functions
Practice Problems
Evaluate each polynomial function
f  x   3 x 2  10
f  1
3  1  10
2
g 3

3 1 10  3 10  7
g  y   6 y 2  11 y  20
6  3  11 3  20  6  9  33  20 
2
54  33  20  87  20  67
5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Monomials by Monomials
Examples:
2
90x
10x  9x 
10

88x
8 x  11x  
3
 5x
7
4
5

x

   5x
5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Monomials by Polynomials
Examples:
4 x  x  4 x  3  4x 16x 12x
3
2
2
8 x  7 x  1  56x 8x
5
4
 5x  3x
3
2
 x  2   15x 5x 10x
5
4
3
5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Two Polynomials
Examples:
2
3
2

5x

10x
x
3x
50x 15
 x  5  x  10 x  3 
2
x 3 15x 2 47x 15
2
4
x
  x  5  3x  4  
12x 3 16x 23x 24x 15x 20 
12x 13x 11x 20
3
2
5.4 – Multiplying Polynomials
Special Products
Multiplying Two Binomials using FOIL
First terms
Outer terms
Inner terms
2
x
x

3
x

4

4x 3x 12 
 

x 7x 12
2
2
x
x

7
x

4

4x 7x 28 



x 2 3x 28
Last terms
5.4 – Multiplying Polynomials
Special Products
Multiplying Two Binomials using FOIL
First terms
Outer terms
Inner terms
2
3
2
2
y

3
y

4

2
y

8y
3y 12 

 
2 y 8y 3y 12
3
y
2
 4 
2
2
y
2
 4  y  4  
2
y 4 y 4 y 16  y 8y 16
4
2
2
4
2
Last terms
5.4 – Multiplying Polynomials
Special Products
Squaring Binomials
2
2
a

b

a

2
ab

b


2
 a  b
2
 a  2ab  b
2
2
2
4
x

5

16x
2  20x  25 


2
16 x  40 x  25
2
 3 x  6 
2
 9x 2 18x  36 
2
9 x 2  36 x  36
5.4 – Multiplying Polynomials
Special Products
Multiplying the Sum and Difference of Two Binomials
2
2
a

b
a

b

a

b



2
2
x
 81
x
x

9
x

9

9x 9x 81 

 
3x  53x  5  9x
2
15x 15x 25  9 x  25
x  8x  8  x  64
2
5 x  135 x  13  25 x  169
2
2
5.4 – Multiplying Polynomials
Special Products
Dividing by a Monomial
ab a b
 
c
c c
where c  0
21x8  9 x 6  12 x 4

3
3x
8
21x
3x3
6
4
9x
12 x
 3  3 
3x
3x
7 x 3x 4x
5
3
6.4 – Dividing Polynomials
Long Division
Review of Long Division
285
5
57
5 285
 25
35
 35
0
783  4
1 95
4 783
4
38
 36
23
 20
3
3
195 
4
3
195
4
6.4 – Dividing Polynomials
Long Division
x 2  12 x  35
 x  5
x 7
x  5 x 2  12 x  35
 x 2  5 x 
7 x  35
 7 x  35
0
 x  5  x  7 
 x 2  12 x  35
6.4 – Dividing Polynomials
Long Division
8x 2  2 x  7
2 x  1
4


3
4x
2x 1
2 x  1 8x 2  2 x  7
2
8
x
   4 x
6x  7
  6x  3 
4
4 

2
4
x

3


8x  2 x  7

2x 1 
2x 1 

6.4 – Dividing Polynomials
Long Division
 2 x 2  15
 x  3
3

 2x  6 x  3
x  3  2 x 2  0 x  15
2


2
x
 6x 

 6x  15
  6x 18 
3
3 

x  3   2 x  6 

x 3

 2 x 2  15
5.7 - Factoring Polynomials
Factoring Trinomials
 x  4 x  5  x
 x  4 x  5 
2
 x  bx  c
2
 5 x  4 x  20  x  9 x  20
2
x  9 x  20
2
20 is the result of the product of 4 and 5.
9 is the result of the sum of 4 and 5.
Factors of 20 are: 1, 20
2, 10
4, 5
5.7 - Factoring Polynomials
Factoring Trinomials
 x  bx  c
2
x  10 x  24
2
Factors of 24 are:
1, 24
x
2, 12
 x
 x  4 x  6

3, 8
4, 6
5.7 - Factoring Polynomials
Factoring Trinomials
 x  bx  c
2
x  27 x  50
2
Factors of 50 are:
1, 50
x
 x
2, 25

 x  2 x  25
5,10
5.7 - Factoring Polynomials
Factoring Trinomials
 x  bx  c
2
4 x  24 x  36
2
4  x  6x  9
2
Factors of 9 are: 1, 9 3, 3
4 x
 x
4  x  3 x  3

5.7 - Factoring Polynomials
Factoring Trinomials
 ax  bx  c
2
2 x  11x  12
2
Factors of 12 are:
1, 12 2,6 3, 4
Factors of 2 are: 1, 2

x

x
 x  4 2 x  3

5.7 - Factoring Polynomials
Factoring Perfect Square Trinomials
x  12 x  36 
x  13 x  36 
2
2

 x

 x  6 x  6
2
x  6
x

x
 x

x  4x  9
Not a perfect square
5.7 - Factoring Polynomials
Factoring Perfect Square Trinomials
x  18 x  81 
2

 x

 x  9 x  9
x
x  9
2
9 x  42 x  49 
x
 x

2

3x  7 3x  7 
3x  7 
2
5.7 - Factoring Polynomials
Factoring the Difference of Two Squares
9s  1 
 3s  13s 1
p  81 
 p  9 p  9
2
2
4 x  100  Not the difference
2
5.7 - Factoring Polynomials
Factoring the Difference of Two Squares
121x  49 y 
2
2
11x  7 y 11x  7 y 
9
3 
3

64c 
  8c    8c  
25 
5 
5
2
5.7 - Factoring Polynomials
Factoring the Sum or Difference of Two Cubes

 a  b a
a  b  a  b  a  ab  b
2
a b
2
3
3
3
3
2
2
 ab  b
x  27
2
x  3   x  3x  9
3




5.7 - Factoring Polynomials
Factoring the Sum or Difference of Two Cubes

 a  b a
a  b  a  b  a  ab  b
2
a b
2
3
3
3
3
2
2
 ab  b
y 8
2
y  2 y  4
3

y2




5.7 - Factoring Polynomials
Factoring the Sum or Difference of Two Cubes

 a  b a
a  b  a  b  a  ab  b
2
a b
2
3
3
3
3
x  64 y
3
x 
2
2
 ab  b
3
4y   x  4 xy 16 y
2

2


5.7 - Factoring Polynomials
Factoring the Sum or Difference of Two Cubes

 a  b a
a  b  a  b  a  ab  b
2
a b
2
3
3
3
3
27 a  b a
2
3
a 27  b
2
a
2
 3  b 9
3
2
2
 ab  b


2

 3b  b
2

End
Of
Presentation
5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Two Polynomials
Examples:
2

x
 10x  5x  50 
x  5x  10
x 2  15x  50
2
12x
4x  53x  4  16 x  15x  20 
12x 2  x  20
5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Two Polynomials
Examples:
 x  3  2 x
2
 5x  4 
2x 5x 4x 6x 15x 12 
3
2
2
2x 3  x 2 11x 12
5.4 – Polynomials and Polynomial Functions
Multiplication
Multiplying Two Polynomials
Examples:
 x  3 y  2
2

 x  3 y  2 x  3 y  2 
x 3xy 2x 3xy 9 y 6 y 2x 6 y 4 
2
2
x 2 6xy 4x 9 y 2 12 y 4