Transcript Polynomials - TCC: Tidewater Community College

Polynomials
By
Dr. Julia Arnold
Tidewater Community College
Copyright 10/19/2002
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,...})
The following are examples of polynomials:
2x4
A one term polynomial is called a monomial.
-5x6 + 7.9x2
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-1 + 2 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.
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
How many terms does the following polynomial have?
7x4 - 3x2 + 4x3 - 9x +5
The polynomial 7x4 - 3x2 + 4x3 - 9x +5 has 5 terms.
1
2
3
4
5
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
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
(2 + 3 + 1) 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.
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 ?
The degree of x7 + 4x8 - 3x9 - 9x3 + 5 is
9
the highest degreed term.
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
(C) 3x2 and 4 x3
(B) -2xyz and 5xyz
The answer is (B) -2xyz is like 5xyz
because
both have the same variables raised to
the same powers or exponents.
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]
Example 1: (2x +3) + (5x - 6)
First remove grouping symbols
2x + 3 + 5x - 6
Next find the like terms
2x + 5x + 3 - 6
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? ]
The distributive property:
a(b + c) = ab + ac
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 - 7 - 9x3 - 8x2 - 7x + 3
Combining like terms
-9x3 - 3x2 + x - 4
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 - 5x2 + 3 + 4x like terms under each other
-8x4 + 3x2 + 2 + 4x
add columns
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)
2x + 3 - 5x + 6
First remove grouping symbols
(Note: Multiplying by -1 causes the
signs to change in your expression.)
2x - 5x and 3 + 6 Add the like terms
-3x + 9 is the result.
Example 2: (8x2 - 2x - 5) - (x3 - 9x2 - 2x + 5)
1(8x2 - 2x - 5) - 1(x3 - 9x2 - 2x + 5)
Remove grouping symbols
8x2 - 2x - 5 - 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
3 - 2x
8x
becomes
-1(5x3 - 3x -10)
-5x3 + 3x + 10
3x3 + x + 10
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.
10 . 3 . x2 . x9 . y1 . y2
= 30x(2+9)y(1+2) = 30x11y3
Example 3: (-4x7y0)(-9x0yz3) =
Write the answer before you click your mouse.
-4 . -9 . x7 . x0 . y0 . y1 . z3
36x(7+0)y(0+1)z3
36x7yz3
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 =
-2x3
+ 6x2
- 18x
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
Example 4: (-3x4 - 5x2 + 1) (-3x2)
Write your answer before you click your mouse.
= -3x4 (-3x2) - 5x2 (-3x2) + 1 (-3x2)
9x6
+ 15x4
- 3x2
Example 5: -2a( a3 + a2 - a + 4) =
Write your answer before you click your mouse.
-2a( a3) + -2a (a2 ) -2a( - a) + -2a(4) =
-2a4
- 2a3
+ 2a2 - 8a
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
O stands for outside.
The outside terms are x and -5
More on the next slide.
(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
L stands for last.
The last terms are 4 and -5
Their product is -20
More on the next slide.
(x + 4)(2x -5)
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
3y(5y) = 15y2
Outside
3y(-6) = - 18y
Inside
-7 (5y) = - 35y
Last
-7 ( -6) = +42
Answer is 15y2 - 18y - 35y + 42
Final Answer is 15y2 - 53y + 42
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
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
3( 4x2 - 3x -2) =
12x2 - 9x -6
Now add down:
8x3 +6x2 - 13x -6
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.
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 - 8 = 3x2 - x + 7 - 8
5x2 5x2 5x2 5x2
5
5x
5x2
Example 2: 9x3 - 4x2 + 8x - 6
3x
Write 9x3 - 4x2 + 18x - 6 = 3x2 - 4x + 6 - 2
3x
3x 3x 3x
3
3x
Division by a Polynomial with 2 or more terms.
Quotient
Divisor
Dividend
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
X2 + 12.
X+3 X2 +0X + 12
2) If there are any missing exponents
in your dividend , you make
space for them by adding a zero
term.
Example 1: Divide (X2 + 5X + 12) by ( X + 3)
Set up the long division
problem.
X+3 X2 + 5X + 12
X
X+3 X2 + 5X + 12
X2 + 3X
X
X+3 X2 + 5X + 12
-X2 - 3X
2X + 12
Divide the first term X2 by the
first term in the divisor, X. Write
the result above 5X.
Multiply X by the divisor X + 3 and
write the answer below the dividend
matching like terms as you go.
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.
We are not finished yet so continue onto the next slide!
X+ 2
X+3 X2 + 5X + 12
-X2 - 3X
2X + 12
- 2X +
6
6
Write the final answer with
the remainder in the form
below.
X+2+ 6
X + 3 X2 + 5X + 12 X+3
-X2 - 3X
2X + 12
-2X - 6
6
Divide the first term 2X by the first
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)
The remainder is 6.
Example 2: Divide (X2 - 5) by ( X - 2)
Set up the long division
Divide the first term X2 by the
problem.
first term in the divisor, X. Write
the result above 0X.
X- 2 X2 + 0X - 5
Multiply X by the divisor X - 2 and
X
write the answer below the dividend
X- 2 X2 + 0X - 5
matching like terms as you go.
X2 - 2x
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
We are not finished yet so continue onto the next page!
X
X- 2 X2 + 0X - 5
-X2 + 2X
+2X - 5
X+2
X- 2
Divide the first term 2X by the first
term X. The answer is 2. Write +2
above the -5.
X2 + 0X - 5
-X2 + 2X
> 2X - 5
- 2X -+ 4
-1
Multiply 2(X - 2) = 2X - 4
Write answer below 2X - 5.
Subtract by changing signs.
The remainder is -1.
X+2+
X - 2 X2 + 0X - 5
-X2 - 2X
2X - 5
-2X + 4
-1
-1
x-2
Write the final answer in the
form on left.
Example 3: Divide 8X3 - 1
2X + 1
Set up the long division problem before you click the mouse.
2X + 1 8X3 + 0X2 + 0X - 1
4X2
2X + 1 8X3 + 0X2 + 0X - 1
4X2
2X+1 8X3 + 0X2 + 0X - 1
8X3 + 4X2
Step 1: Divide 8X3 by 2X
Write answer before you click.
Step2: Multiply 4X2 by
divisor. Write answer before
you click.
We are not finished yet so continue onto the next page!
4X2
2X + 1 8X3 + 0X2 + 0X - 1
- 8X3 +
- 4X2
- 4X2 + 0X - 1
4X2 - 2X
2X+1
8X3 + 0X2
-8X3 - 4X2
- 4X2
+-4X2
+ 0X - 1
+ 0X - 1
+- 2X
Step 3: Subtract by changing
signs and bring down left
over terms.
Repeat Steps 1 - 3 again
Step 1: Divide - 4X2 by 2X
Step2: Multiply -2X by
divisor 2x + 1.
Step3: Subtract
4X2 - 2X
2X+1
2X+1
8X3 + 0X2 + 0X - 1
-8X3 - 4X2
- 4X2 + 0X - 1
+ 2 -+ 2x
-4x
2x - 1
Step 3: Subtract by changing
signs and bring down left
over terms.
Repeat Steps 1 - 3 again
Step 1: Divide 2X by 2X
4X2 - 2X + 1 + -2
2x +1
8X3 + 0X2 + 0X - 1
-8X3 - 4X2
- 4X2 + 0X - 1
+ 4x2 + 2x
2x -1
- 2x +- 1
-2
Step2: Multiply 1 by
divisor.
Step 3: Subtract
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