This document covers exponents, polynomials, and operations involving polynomials such as addition, subtraction, multiplication, and division. It defines polynomials and describes how they can be simplified using properties of exponents. It provides examples of multiplying and dividing polynomials using standard algorithms like distributing terms and cancelling out common factors. Special cases for multiplying binomials are also discussed.

1. 0.3 Polynomials
Chapter 0 Review of Basic Concepts

2. Concepts & Objectives
⚫ Exponents
⚫ Simplify exponential expressions
⚫ Polynomials
⚫ Add, subtract, multiply, and divide polynomial
expressions

3. Properties of Exponents
⚫ Recall that for variables x and y and integers a and b:
Multiplying
Dividing
Power to a Power
Distributing a Power
Zero Exponent
+
=a b a b
x x x
−
=
a
a b
b
x
x
x
( ) =
b
a ab
x x
( ) =
a a a
xy x y
=0
1x

4. Simplifying Exponents
⚫ Example: Simplify
⚫ 1.
⚫ 2.
⚫ 3.
−3 2 2
2 5
25
5
x y z
xy z
( )−
4
2 3
2r s t
( )
4
2 5
3x y−
−

5. Simplifying Exponents
⚫ Example: Simplify
⚫ 1.
⚫ 2.
⚫ 3.
−3 2 2
2 5
25
5
x y z
xy z
( )−
4
2 3
2r s t
( )
4
2 5
3x y−
−
3 1 2 2 2 525
5
x y z− − − −
=
( ) ( ) ( )4 2 4 3 4 14
2 r s t
−
=
( ) ( ) ( )4 2 4 5 4
3 x y
−
= −
− −
= 2 4 3
5x y z
−
= 8 12 4
16r s t
8 20
81x y−
=

6. Polynomials
⚫ A polynomial is defined as a term or a finite sum of
terms, with only positive or zero integer exponents
permitted on the variables.
⚫ The degree of a term with one variable is the largest
exponent in the expression. The degree of a term
containing more than one variable has degree equal to
the sum of all of the variables’ exponents.
⚫ The degree of the polynomial is the greatest degree of
any term in the polynomial.

7. Polynomials (cont.)
⚫ A polynomial containing exactly three term is called a
trinomial. If it contains exactly two terms, it is called a
binomial, and a single-term polynomial is called a
monomial.
⚫ Since the variables used in polynomials represent real
numbers, a polynomial represents a real number. This
means that all of the properties of real numbers hold for
polynomials.

8. Polynomials (cont.)
⚫ Using the distributive property, we can simplify the
following:
3m5 ‒ 7m5 + 2m2
3 ‒ 7m5 + 2m2
‒4m5 + 2m2
⚫ Thus, polynomials are added/subtracted by adding or
subtracting coefficients of like terms.
⚫ Polynomials in one variable are usually written with
their terms in descending order of degree.

9. Multiplying Polynomials
⚫ To multiply polynomials, multiply each term by all of the
other terms.
Example: ( )( )
( )( ) ( )( ) ( )( )
( )( ) ( )( ) ( )( )
2
2
2
3 2 2
3 2
2 3 5
2 3 5
2 3 5
6 9 15 8 12 20
6 17 27 20
3
3 3
4
4 4 4
3
x x
x x
x x
x x x x x
x x
x
x
x
x x
− +
= + − +
+ + − +
= − + − + −
−
− +
−
−
−
−
=

10. Multiplying Polynomials
⚫ You can also use tools such as the “box” method:
2x2 ‒3x 5
3x
‒4

11. Multiplying Polynomials
⚫ You can also use tools such as the “box” method:
2x2 ‒3x 5
3x 6x3 ‒9x2 15x
‒4

12. Multiplying Polynomials
⚫ You can also use tools such as the “box” method:
2x2 ‒3x 5
3x 6x3 ‒9x2 15x
‒4 ‒8x2 12x ‒20

13. Multiplying Polynomials
⚫ You can also use tools such as the “box” method:
Combine like terms:
2x2 ‒3x 5
3x 6x3 ‒9x2 15x
‒4 ‒8x2 12x ‒20
3 2
6 17 27 20x x x− + −

14. Multiplying Polynomials
⚫ There are a couple of special patterns to be aware of
when multiplying two binomials:
⚫ Product of Sum and Difference/Difference of Squares
⚫ Square of a Binomial
( )( ) 2 2
a b a b a b+ − = −
( ) = +
2 2 2
2a b a ab b

15. Multiplying Polynomials
Examples: Simplify the following
⚫
⚫
( )( )2 3 2 3x x+ −
( )
2
7 5x −

16. Multiplying Polynomials
Examples: Simplify the following
⚫
⚫
( )( ) ( )+ − = −
= −
2 2
2
2 3 2 3 2 3
4 9
x x x
x
( ) ( ) ( )( )− = − +
= − +
2 2 2
2
7 5 7 2 7 5 5
49 70 25
x x x
x x

17. Multiplying Polynomials
EVERY TIME YOU DO THIS:
A KITTEN DIES
( )
2 2 2
x y x y+ = +
Fair Warning!

18. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1m m m m− + + −

19. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1m m m m− + + −
2m‒1  4m3 ‒ 8m2 + 4m + 6

20. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1m m m m− + + −
2m2
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 – 2m2
You have to cancel
out the first term,
so this has to be
the same.

21. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1m m m m− + + −
2m2
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
–Now change all of
the signs.

22. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1m m m m− + + −
2m2 ‒ 3m
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
–6m2 + 3m
–

23. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1m m m m− + + −
2m2 ‒ 3m
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
+6m2 – 3m
m + 6
–

24. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1m m m m− + + −
2m2 ‒ 3m + ½
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
+6m2 – 3m
m + 6
m – ½
–

25. Dividing Polynomials
⚫ To divide polynomials, we use much the same process
we use to divide whole numbers:
Example: Divide 3 2
4 8 4 6 by 2 1m m m m− + + −
2m2 ‒ 3m + ½
2m‒1  4m3 ‒ 8m2 + 4m + 6
4m3 + 2m2
‒6m2 + 4m
+6m2 – 3m
m + 6
m + ½
13
2
–
–
12
6
2
 
= 
 

26. Dividing Polynomials
⚫ Therefore,
⚫ If either polynomial does not have a term for each
power, you will need to insert a placeholder for it.
3 2
2
13
4 8 4 6 1 22 3
2 1 2 2 1
m m m
m m
m m
− + +
= − + +
− −

27. Dividing Polynomials
⚫ Example: Divide 3x3 ‒ 2x2 ‒ 150 by x2 ‒ 4
x2 + 0x ‒ 4  3x3 ‒ 2x2 + 0x ‒ 150

28. Dividing Polynomials
⚫ Example: Divide 3x3 ‒ 2x2 ‒ 150 by x2 ‒ 4
3x
x2 + 0x ‒ 4  3x3 ‒ 2x2 + 0x ‒ 150
3x3 + 0x2 ‒ 12x
‒ 2x2 + 12x ‒ 150

29. Dividing Polynomials
⚫ Example: Divide 3x3 ‒ 2x2 ‒ 150 by x2 ‒ 4
3x ‒ 2
x2 + 0x ‒ 4  3x3 ‒ 2x2 + 0x ‒ 150
3x3 + 0x2 ‒ 12x
‒ 2x2 + 12x ‒ 150
‒ 2x2 + 0x + 8
12x ‒ 158

30. Dividing Polynomials
⚫ Example: Divide 3x3 ‒ 2x2 ‒ 150 by x2 ‒ 4
So,
3x ‒ 2
x2 + 0x ‒ 4  3x3 ‒ 2x2 + 0x ‒ 150
3x3 + 0x2 ‒ 12x
‒ 2x2 + 12x ‒ 150
‒ 2x2 + 0x + 8
12x ‒ 158
3 2
2 2
3 2 150 12 158
3 2
4 4
x x x
x
x x
− − −
= − +
− −

31. Classwork
⚫ College Algebra
⚫ 0.3 Assignment – page 30: 12-28, page 19: 44-64 (x4),
page 7: 66-90
⚫ 0.3 Classwork Check
⚫ Quiz 0.2