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SS 4.1 Exponents and Polynomials
Definition of an exponent
an =
Definition of a negative exponent
a-n =
Definition of zero exponent if a  0
a0 =
Property #1 of exponents
am an =
Property #2 of exponents
(am)n =
Property #3 of exponents
(ab)m =
or
a m =
b
Property #4 of exponents
am =
an
Example:
32
Example:
x2
Example:
x2 x3 =
Example:
(a2 b)(ab) =
Example:
(a3)2 =
81
Example:
(24)2 =
Example:
a
b
Example:
| a2 |3 =
| y3 |
Example:
(ab)2 =
Example:
(2a)2 =
Example:
a2 b =
a b2
Example:
4 a4 =
2 a3
Example:
(x2 y)2 =
Example:
(x2 y)3 (xy)
Example:
(x2 y)2 (xy)2 =
x2y
Example:
7 x0 + (-x0) =
2
=
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SS 4.2 Adding and Subtracting Polynomials
Definitions
polynomial –
monomial –
binomial –
trinomial –
constant –
term –
coefficient –
degree of a term –
degree of a polynomial –
ordering a polynomial –
Example:
a)
b)
c)
d)
What is the degree of the term?
b2
2 b3
x2 y2z
1
Example:
a)
b)
c)
What is the degree of the polynomial?
a2 + 3a + 5
3a + 4 a3  2a2  6
ab2 + 3a2b3 + 2a2  4
Recall that simplifying meant:
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Addition and subtraction of polynomials is simplification!!
Steps: 1)
2)
3)
Remove grouping symbols
Group Like Terms
Combine Like Terms
Example:
(8x2 + 2x + 5) + (x2 + 5x + 3)
Step 1
Step 2
Step3
Example:
[(8x2 + 2) + (-7x + 3)] + (x2 + 3)
Step 1
Step 2
Step 3
Example:
(7x2  2x + 3)  (5x2  4)
Step 1
Step 2
Step 3
There is another way to think about adding and subtracting polynomials.
This is columnar addition and subtraction. We must really focus on
ordering the polynomial to do this.
Steps: 1)
2)
3)
4)
Order polynomials being added or subtracted
Remove subtraction
Stack in columns
Add
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Example:
(7x  2x2 + 3)  (5 + x2  2x)
Order and leave blanks
for missing degrees
Remove subtraction
Stack in columns
Add
Example:
(9x2  9) + (x2 + x + 7)
Order and leave blanks
for missing degrees
Remove subtraction
Stack in columns
Add
Example:
Subtract (x2  9) from (x2 + 2 x  3)
Create Problem
Remove Subtraction
Order & Leave Blanks
Stack Like Terms
Add
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Evaluating a polynomial is just like evaluating any algebraic expression.
Steps:
1) Leave blanks where variables are
2) Fill the blanks with the value of the variable
3) Solve the resulting numeric expression
Example:
(x2  3x + 3) if x = -1
Leave blanks
Fill blanks
Expand
Simplify
Example:
(x2y + 3x + 2y  5) if x = 4 and y = -3
Leave blanks
Fill blanks
Expand
Simplify
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SS 4.3 Multiplying Polynomials
Multiplying polynomials is an application of the distributive property. This
is also called expanding.
Review
a(b + c) =
Monomial x Polynomial
Example:
2x (x2 + 2x + 3)
Distributive
Exponent Rules
Example:
-4x2 (x  2x2 + 3)
Distributive
Exponent Rules
Binomial x Binomial
Now we'll extend the distributive property further and to help us remember
how we will have an acronym called the FOIL method.
(a + b)(d + c)
F
O
I
L
Example:
(x + 2) (x + 3)
FOIL
Exponents
Combine Like Terms
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Example: (x  5) (2x + 3)
FOIL
Exponents
Combine Like Terms
Example:
(x2 + y) (x  y)
FOIL
Exponents
Combine Like Terms
Polynomial x Polynomial
Example:
(2x + 3) (x2 + 4x + 5)
Distributive
Exponents
Combine Like
Example:
(x2 + 2x  7) (x2  2x + 1)
Distributive
Exponents
Combine Like Terms
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Example:
(x + 1)2 (2x + 3)
Expand
Left to Rt. Distributive
Exponents
Simplify
Distributive Again
Exponents Again
Simplify Again
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SS 4.4 Special Cases
Square of a Binomial
(a + b)2 =
(a  b)2 =
Multiplying the sum and difference of 2 terms
(a + b) (a  b) =
These are very important for the next chapter so take notice of the
polynomials and their expansions.
Example:
(2x + 3)2 =
Form
Exponents
Example:
(-2a + b)2 =
Form
Exponents
Example:
(7a  2)2 =
Form
Exponents
Example:
(a + 2)(a  2) =
Form
Exponents
Example:
(2x  4) (2x + 4) =
Form
Exponents
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SS 4.5 Negative Exponents and Scientific Notation
We've already discussed negative exponents, but let's talk about them again.
a-1 = the reciprocal of a
a-n = (a-1)n = | 1 | n = 1
|a |
an
or = (an)-1 = 1
an
Example:
3 -1 =
Example:
3 -2 =
Example:
x -5 =
Example:
x4 =
x7
Example:
1 =
2z -2
Example:
| x -1 y 2 | -2 =
| y -3 x -3 |
Example:
|x 2 |-2 =
| x -1 y -2 |
Example:
4 -1 + | 4 |-1 =
|3 |
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Scientific Notation
When we use 10 as a factor 2 times, the product is 100.
102 = 10 x 10 = 100
second power of 10
When we use 10 as a factor 3 times, the product is 1000.
103 = 10 x 10 x 10 = 1000
third power of 10.
When we use 10 as a factor 4 times, the product is 10,000.
104 = 10 x 10 x 10 x 10 = 10,000
fourth power of 10.
From this, we can see that the number of zeros in each product equals the
number of times 10 is used as a factor. The number is called a power of 10.
Thus, the number
100,000,000
has eight 0's and must be the eighth power of 10. This is the product we get
if 10 is used as a factor eight times!
Recall earlier that we learned that when multiplying any number by powers
of ten that we move the decimal to the right the same number of times as
the number of zeros in the power of ten!
Example : 1.45 x 10 = 14.5
Recall also that we learned that when dividing any number by powers of ten
that we move the decimal to the left the same number of times as the
number of zeros in the power of ten!
Example : 547.92  100 = 5.4792
Because we now have a special way to write powers of 10 we can write the
above two examples in a special way -- it is called scientific notation .
Example : 1.45 x 101 = 14.5 ( since 101 = 10 )
Steps To Writing a Number in Scientific Notation:
1) Put the decimal just to the right of the first digit that isn't zero.
2) Multiply this number by 10x ( x is a whole number ) to tell your
reader where the decimal point is really located. The x tells your
reader how many zeros you took away! (If it is a number that is 1
or greater, then the x will be positive, otherwise the x will be
negative.)
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Example : Change 17,400 to scientific notation.
1) Decimal 1 7 4 0 0
2) Multiply
x 10
Example : Change 8450 to scientific notation.
1) Decimal 8 4 5 0
2) Multiply
x 10
Example : Change 104,750,000 to scientific notation.
1) Decimal 1 0 4 7 5 0 0 0 0
2) Multiply
x 10
Now, you may be asking yourself, scientific notation does a great job of
showing me to move the decimal to the right and thus multiplication -- but,
how do I show moving the decimal to the left and thus division? The
answer is still scientific notation, but this time we will use negative
exponents, because as you may recall -- a power of negative one means
taking the reciprocal of a number, and thus dividing by that number!!
Example :
547.92 x 10-2 = 5.4792 ( since 102 = 100 and
[ 102 ]-1 = 1 which means
100
divided by 100)
Example : Change 6.259 x 10-3 to standard form.
1) Move Decimal Left ____ times
[ standard form means -- written as a real number ]
Example :
Change 7.193 x 105 to standard form
1) Move Decimal to the Right ________ times.
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Example : Write 0.00902 in scientific notation.
1) Decimal 0 0 0 9 0 2
2) Multiply
x 10
Example : Write 0.00007200 in scientific notation
1) Decimal 0 0 0 0 7 2 0 0
2) Multiply
x 10
Example : Write 0.92728 in scientific notation.
1) Decimal 9 2 7 2 8
2) Multiply
x 10
** Note: When a number is written correctly in scientific notation, there is
only one number to the left of the decimal. Scientific notation is always
written as follows: a x 10x, where a is a described above and x is an
integer.
Multiplying with Scientific Notation
What happens if we wish to do the following problem,
7 x 102 x 103
We can think of 102 and 103 as "decimal point movers." The 102 moves
the decimal two places to the right and then the 103 moves the decimal three
more places to the right. When we are finished we have moved the decimal
five places to the right.
Steps for Multiplying with Scientific Notation:
1) Multiply the whole numbers
2) Add the exponents of the "decimal point movers"
3) Rewrite in scientific notation where there is only one place value
to the left of the decimal
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Example : Multiply (3 x 102 ) ( 2 x 104)
1) Multiply
2) Add exponents
3) Rewrite
Example : Multiply (2 x 10-2 ) (3 x 106)
1) Multiply
2) Add exponents
3) Rewrite
Example : Multiply (1.2 x 103 ) (12 x 107)
1) Multiply
2) Add exponents
3) Rewrite
Example : Multiply (9 x 107 ) (8 x 10-3)
1) Multiply
2) Add exponents
3) Rewrite
Steps for Dividing with Scientific Notation:
1) Divide the whole numbers
2) Subtract the exponents of the "decimal point movers"
3) Rewrite in scientific notation where there is only one place value
to the left of the decimal
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Example:
( 9 x 105 ) =
( 3 x 102 )
1) Divide
2) Subtract exponents
3) Rewrite
Example:
( 2.5 x 107 ) =
( 2.5 x 105 )
1) Divide
2) Subtract exponents
3) Rewrite
Example:
( 2 x 10 -2 ) =
( 1.5 x 105 )
1) Divide
2) Subtract exponents
3) Rewrite
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SS 4.6 Division of Polynomials
Before we begin this section let's recall some things about fractions
and division:
Fractions
1 + 2 = 1 + 2
17
17
17
3x + 1 = 3x + 1
7
7
7
Division
25 2552
Recall that we start placing our numbers
over the last digit of the whole number in the
dividend that the divisor will go into, then
we multiply that number by the divisor
subtract and bring down the next number
until we run out of numbers to bring down.
If there are remainders then we put the
remainder over the divisor to create a
fraction, which leads us to the next point …
Mixed Numbers
3 1 =
3 + 1
2
2
Checking Division
Let's take the answer from the division problem above and
review how to check…
1) Multiply whole numbers
2) Add remainder
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Division of a Polynomial by a Monomial
1) Break down as sum of fractions
2) Use exponent rule of division to simplify each term
Example:
x2 + 3x =
x
Break down into sums
Exponent rules
Example:
15 x3 y + 3 x2 y  3 y =
xy
Break down into sums
Exponent rules
Example:
27 x5  3 x3 + 4x =
9 x2
Break down into sums
Exponent rules
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Division of Polynomial by Binomial
1) Order polynomial, leaving blanks for missing degreed terms
2) Write as a division problem
3) Divide 1st term of dividend by 1st term of divisor
4) Multiply quotient in 3 by divisor and subtract
5) Bring down next term
6) Repeat steps 3-5 until the degree of the remainder is less than the
degree of the divisor polynomial
7) Write remainder as a fraction added to quotient polynomial
8) Check
Example:
x2 + 2x + 4 =
x + 2
Example:
x3  2x + 21 =
x + 3
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Example:
4x2  2x + 1 =
x  1
Example:
2x2  x + 1 =
3x  1
Now it's your turn
Example:
x2  x  2 =
x + 1
100
Example:
x3 + 2x  1 =
x + 4
** Note: A polynomial can be divided by a monomial using long division,
but why make it so complicated!
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