1. By Dr. Julia Arnold Tidewater Community College
Polynomials By
Dr. Julia Arnold
Tidewater Community College
Copyright 10/19/2002

2. Introduction What are Polynomials?
A polynomial in x consists of a finite number of terms of the form axn where a can be any Real number but n must be a whole number. (Recall a term is any algebraic expression separated from another algebraic expression by “+” or “-” signs. Whole numbers are {0,1,2,3,4,...})

3. The following are examples of polynomials:
2x A one term polynomial is called a monomial.
-5x x2 A two term polynomial is called a binomial.
2x2 + 3x - 1 A three term polynomial is called a trinomial.
The following are not polynomials:
3x-2- 4x is not a polynomial because the exponents on the variables are not whole numbers.
19y1/2 + 5 is not a polynomial because the exponent on the variable is not a whole number.

4. The following are examples of polynomials in more than one variable:
2x4 y2 is a monomial in x and y
-5x6yz + 7.9x2yz2 is a binomial in x, y & z
2x2w + 3xw2 - w3 is a trinomial in x and w

5. How many terms does the following polynomial have?
7x4 - 3x2 + 4x3 - 9x +5
The polynomial 7x4 - 3x2 + 4x3 - 9x +5 has 5 terms.

6. Descending Powers
Writing a polynomial in descending powers means to begin with the term having the largest exponent on the variable and then proceeding to the lowest.
For example: - 3x2 + 4x3 - 9x + x4 +5 would
be written x4 + 4x3 - 3x2 - 9x + 5

7. Degree of a Term
The degree of a term is the sum of the exponents on all variables.
For example: the degree of 5x2y3z is
( ) or 6
For the polynomial x4 + 4x3 - 3x2 - 9x + 5 the degree of each term from left to right is
4, 3, 2, 1, and 0.
The constant 5 is equal to 5x0, thus it has degree 0.

8. Degree of a Polynomial
The degree of a polynomial is the largest degree of any one term. Thus in the preceding polynomial,
x4 + 4x3 - 3x2 - 9x + 5, the degree would be 4.
What is the degree of
x7 + 4x8 - 3x9 - 9x3 + 5 ?

9. The degree of x7 + 4x8 - 3x9 - 9x3 + 5 is 9
the highest degreed term.

10. What are Like Terms?
Like terms are terms with the same variables raised to the same powers.
For example:
5x2y3 is like -4y3 x2 but is not like 5y3x
x is like .35x but is not like x2
Which of the following pairs are pairs of like terms?
(A) 3xy and 2yz (B) -2xyz and 5xyz
(C) 3x2 and 4 x3

11. The answer is (B) -2xyz is like 5xyz because
both have the same variables raised to the same powers or exponents.

12. Adding Polynomials To add polynomials (1) remove the grouping symbols,
(2) find the like terms of the polynomial, and then
(3) add the numerical coefficients of the like terms.
[Note: the numerical coefficient is the number with the variables; i.e. 3xyz has numerical coefficient 3, -5x2 has numerical coefficient -5, and x has numerical coefficient 1]

13. Example 1: (2x +3) + (5x - 6)
First remove grouping symbols
2x x - 6
Next find the like terms
2x + 5x
Add numerical coefficients
(2+5)x + (3 - 6) = 7x - 3
[Question: Does 2x + 5x = (2 + 5)x remind you of a property stressed earlier in the course? ]

14. The distributive property:
a(b + c) = ab + ac

15. Example 2:
(5x2 + 8x - 7) + (-9x3 - 8x2 - 7x + 3) [Remember to remove the grouping symbols, multiply by whatever number is in front of the grouping symbols, using the distributive property. That number in this problem is 1 for both polynomials. 1(5x2 + 8x - 7) + 1(-9x3 - 8x2 - 7x + 3)]
5x2 + 8x x3 - 8x2 - 7x + 3
Combining like terms
-9x3 - 3x2 + x - 4

16. Adding in Columns
If you prefer, you can use the column method of adding polynomials.
Like terms are placed under each other.
Example 3:
Add
(-10x4 + 8x2 - 1) and ( 2x4 - 5x2 + 4x + 3)
Write
-10x4 + 8x2 - 1
2x4 - 5x x like terms under each other
-8x4 + 3x x add columns

17. Subtracting Polynomials
To subtract polynomials you must remove the
grouping symbols by multiplying the first
expression by 1 and the second expression by -1.
Example 1:
(2x +3) - (5x - 6)
1(2x + 3) -1 ( 5x - 6) First remove grouping symbols x x + 6 (Note: Multiplying by -1 causes the signs to change in your expression.)
2x - 5x and Add the like terms
-3x is the result.

18. Subtracting in Columns
Example 2: (8x2 - 2x - 5) - (x3 - 9x2 - 2x + 5)
1(8x2 - 2x - 5) - 1(x3 - 9x2 - 2x + 5)
Remove grouping symbols
8x2 - 2x x3 + 9x2 + 2x - 5
Add like terms
- x3 + 17x2 -10 (the answer)
Subtracting in Columns
Be very careful when using this method. You must make sure you change all the signs of the polynomial being subtracted.
Example 3: Subtract 5x3 - 3x -10 from 8x3 - 2x
8x3 - 2x
8x3 - 2x
-5x3 + 3x + 10
becomes
-1(5x3 - 3x -10)
3x3 + x

19. Multiplication of Polynomials
Multiplying a Monomial by a Monomial
Example 1: (-2x6)(3x4) = -2
. 3
. x6
. x4
= -6x(6+4)
= -6x10
Example 2: (10x2y)(3x9y2)
Write the answer before you click your mouse.
x2 . x9 . y1 . y2
= 30x(2+9)y(1+2) = 30x11y3

20. Example 3: (-4x7y0)(-9x0yz3) =
Write the answer before you click your mouse.
x7 . x0 . y0 . y1 . z3
36x(7+0)y(0+1)z3
36x7yz3

21. Multiplying a Monomial by a Polynomial
Example 1: Distribute -2x thru parenthesis to each term
-2x ( x2 - 3x + 9) = -2x(x2) - (-2x)3x + (-2x)9 =
-2x x x
Example 2: 3a2(-2a3 + 8a - 10) =
Write your answer before you click your mouse.
3a2(-2a3) + 3a2(8a) + 3a2(-10) = -6a5 + 24a3 - 30a2
Example 3: (5x2 - 4x + 6) (3x) =
Write your answer before you click your mouse.
5x2 (3x) - 4x (3x) + 6 (3x) = 15x3 - 12x2 +18x

22. Example 4: (-3x4 - 5x2 + 1) (-3x2)
Write your answer before you click your mouse.
= -3x4 (-3x2) - 5x2 (-3x2) + 1 (-3x2)
9x x x2
Example 5: -2a( a3 + a2 - a + 4) =
Write your answer before you click your mouse.
-2a( a3) + -2a (a2 ) -2a( - a) + -2a(4) =
-2a a a a

23. F stands for first. In the problem (x + 4)(2x -5)
Multiplying a Binomial by a Binomial
To multiply two binomials together we use an acronym
called FOIL to help us remember the products.
F stands for first. In the problem (x + 4)(2x -5)
The first terms are x and 2x
Their product is 2x2
First
O stands for outside (x + 4)(2x -5)
The outside terms are x and -5
Outside
More on the next slide.

24. I stands for inside. (x + 4)(2x -5)
Multiplying a Binomial by a Binomial Continued
FOIL stands for First, Outside, Inside, Last.
I stands for inside (x + 4)(2x -5)
The inside terms are 4 and 2x
Their product is 8x
Inside
L stands for last (x + 4)(2x -5)
The last terms are 4 and -5
Their product is -20
Last
More on the next slide.

25. Putting all the products together we get:
(x + 4)( 2x - 5) = 2x2 - 5x + 8x - 20
F O I L
Combining like terms the final answer is
2x2 + 3x - 20
Example 1: Multiply (3y - 7)(5y - 6)
First y(5y) = 15y2
Outside y(-6) = - 18y
Inside (5y) = - 35y
Last ( -6) = +42
Answer is 15y2 - 18y - 35y + 42
Final Answer is 15y2 - 53y + 42

26. Example 2: Multiply (a + b)(c + d) + ad
Do you see that each term of the first polynomial is multiplied by each term in the second polynomial?
Example 2: Multiply (a + b)(c + d)
Distribute a thru (c + d) a(c + d) = ac
+ ad
First
Outside
Then distribute b b(c + d) = bc
+ bd
Inside
Last
Final Answer is ac + ad + bc + bd

27. Example 3: Multiply (2x + 3)( 4x2 - 3x -2)
If you understand this basic premise: that each term of the first polynomial is multiplied by each term in the second polynomial, then it will be an easy transition to multiply polynomials containing more than two terms.
Example 3: Multiply (2x + 3)( 4x2 - 3x -2)
Because the second polynomial is not a binomial
we cannot use FOIL. Instead multiply 2x by
( 4x2 - 3x -2) and then multiply 3 by ( 4x2 - 3x -2).
The result is 2x( 4x2 - 3x - 2) =
8x3 - 6x2 - 4x
then ( 4x2 - 3x -2) =
12x2 - 9x -6
Now add down: x3 +6x2 - 13x -6

28. Division of Polynomials
There are two types of division techniques. The first kind that will be illustrated is division by a monomial.
The second kind is for division by any other type of polynomial.
Occasionally, monomial division produces some unexpected answers. If you try to use the “second method” for dividing by a monomial, you may find yourself unable to complete the task.

29. Division by a Monomial Divisor
Example 1: Divide (3x4 - 5x3 +7x - 8) by 5x2
Write each term of the dividend as a fraction with a
denominator of 5x2.
Simplify each fraction to...
3x4 - 5x3 + 7x = 3x2 - x
5x2 5x x x x x2
Example 2: 9x3 - 4x2 + 8x - 6 3x
Write 9x3 - 4x x = 3x2 - 4x
3x x 3x 3x x

30. Division by a Polynomial with 2 or more terms.
Divide (12 + X2 ) by (X + 3)
In a long division problem you
must follow two set-up rules.
1) The dividend must be arranged
in descending powers. Thus
12 + X2 must be written as X
2) If there are any missing exponents
in your dividend , you make
space for them by adding a zero
term.
Quotient
Divisor
Dividend
X+3
X2 +0X + 12

31. Example 1: Divide (X2 + 5X + 12) by ( X + 3)
Set up the long division
problem.
Divide the first term X2 by the
first term in the divisor, X. Write
the result above 5X.
X+3
X2 + 5X + 12
Multiply X by the divisor X + 3 and
write the answer below the dividend
matching like terms as you go.
X2 + 5X + 12
X2 + 3X
X+3 X
Subtract the bottom line by
changing the signs of the bottom
line you just wrote. When finished
bring down the next term, which is
12.
X2 + 5X + 12
-X2 - 3X
2X + 12
X+3 X
We are not finished yet so continue onto the next slide!

32. Good Job! Divide the first term 2X by the first X
term X. The answer is 2. Write 2
above the 12. (click mouse)
Multiply 2(X + 3) = 2X + 6
Write answer below 2X + 12.
(click mouse)
Subtract by changing signs.
(Click mouse twice)
X2 + 5X + 12
-X2 - 3X
2X + 12
X+3 X
+ 2 - -
2X + 6 6
Write the final answer with the remainder in the form below.
The remainder is 6.
X2 + 5X + 12
-X2 - 3X
2X + 12
-2X 6
X + 3 X
X+3
Good Job!

33. Example 2: Divide (X2 - 5) by ( X - 2)
Set up the long division
problem.
Divide the first term X2 by the
first term in the divisor, X. Write
the result above 0X.
X- 2
X2 + 0X - 5
Multiply X by the divisor X - 2 and
write the answer below the dividend
matching like terms as you go.
X2 + 0X - 5
X x
X- 2 X
Subtract the bottom line by
changing the signs of the bottom
line you just wrote. When finished
bring down the next term, which is -5
X2 + 0X - 5
-X2 + 2X
+2X - 5
X- 2 X
We are not finished yet so continue onto the next page!

34. X2 + 0X - 5
-X2 + 2X
> 2X - 5
2X - 4
X- 2
X + 2
Divide the first term 2X by the first
term X. The answer is 2. Write +2
above the -5.
Multiply 2(X - 2) = 2X - 4
Write answer below 2X - 5.
Subtract by changing signs.
The remainder is -1. - + -1
X2 + 0X - 5
-X2 - 2X
2X - 5
-2X + 4 -1
X - 2 X
x - 2
Write the final answer in the
form on left.

35. Example 3: Divide 8X3 - 1
2X + 1
Set up the long division problem before you click the mouse.
8X3 + 0X2 + 0X - 1
2X + 1
Step 1: Divide 8X3 by 2X
Write answer before you click.
2X + 1
4X2
8X3 + 0X2 + 0X - 1
Step2: Multiply 4X2 by
divisor. Write answer before you click.
8X3 + 0X2 + 0X - 1
8X3 + 4X2
2X+1
4X2
We are not finished yet so continue onto the next page!

36. 4X2
2X + 1
8X3 + 0X2 + 0X - 1
8X X2
Step 3: Subtract by changing
signs and bring down left
over terms. - -
- 4X2 + 0X - 1
Repeat Steps again
Step 1: Divide - 4X2 by 2X
8X3 + 0X2 + 0X - 1
-8X3 - 4X2
- 4X2 + 0X - 1
2X+1
4X2 - 2X
Step2: Multiply -2X by
divisor 2x + 1. + +
-4X2 - 2X
Step3: Subtract

37. +
8X3 + 0X2 + 0X - 1
-8X3 - 4X2
- 4X2 + 0X - 1
2X+1
4X2 - 2X
-4x2 - 2x
Step 3: Subtract by changing
signs and bring down left
over terms.
2x - 1
Repeat Steps again
Step 1: Divide 2X by 2X +
8X3 + 0X2 + 0X - 1
-8X3 - 4X2
- 4X2 + 0X - 1
2X+1
4X2 - 2X
4x2 2x
+ 1
+ -2
2x +1
Step2: Multiply 1 by
divisor.
Step 3: Subtract
2x -1 - -
Go To Practice Problems
2x + 1 -2