Download Getting Ready for Chapter 5 – Polynomials and Polynomial Functions

Getting Ready for Chapter 5 – Polynomials and Polynomial Functions
If a is a real number and n is a positive integer, then a n means a multiplied by
itself, n times.
25  2  2  2  2  2  32
(3)4  (3)(3)(3)(3)  81
In the expression 5x 2
x is the base
2 is the exponent
5 is the coefficient of x 2
In the expression (6 x)3
6x is the base
3 is the exponent
6 is the coefficient of x
If you are combining 2 exponential expressions with the same base raised to
the same power, you combine the coefficients and copy everything else.
5x4  2 x4  7 x4
You can’t combine 2 exponential expressions with different bases, or with
bases raised to different powers, because they are not like terms!
If you are multiplying 2 exponential expressions with the same base, you
multiply the coefficients, copy the base and add the exponents.
5 x 2  7 x 4  5  7  x  x  x  x  x  x  35 x 6
If you are dividing 2 exponential expressions, the rule deals with subtraction.
However, subtraction is not commutative so if you do it backwards you get the
wrong answer. Therefore, I use “Love and Marriage”. Consider the problem:
x7
x4
That’s really
xxxxxxx xxxxxxx xxx


 x3
xxxx
xxxx
1
An easy way to think of it is “Love and Marriage”:
There are 7 x’s living in the attic apartment and 4 x’s living in the basement. If
the 4 x’s in the basement fall in love with 4 of the x’s in the attic and they get
married and move out, how many x’s are left where?
3 in the attic!
That way it’s easy to solve complicated problems without even thinking about
it! If negatives are involved, determine the sign of the answer first. Just
remember that coefficients are actual numbers, so if it says to divide, you need
to divide them!
2 x6 y 2 z 9
 z8

12 x8 y5 z 6 x 2 y 3
Any number to the zero power equals 1.
Why is that? We know
x 0 x5  x 05  x5
And we know that the identity element for multiplication is 1. We just got
identically the same thing back by multiplying by x to the zero; therefore x to
the 0 must equal 1. We could do the same thing with any base, so anything to
the zero power equals 1.
Negative exponents tell you that something is in the wrong position in a
fraction.
1
x3
3


x
x 3 1
x 2 
1
x2
Let’s look at some examples:
52 
1
1

52 25
1 6 1
6
6x  6  2   2  2
x
1 x
x
2
6 0   1
The power rule tells us that when we raise a power to a power, we multiply
the exponents.
 2 x    2 x  2 x  2 x   8x
4 3
4
4
4
12
Operation
Raise to a power
Multiply or divide
Add or subtract
Coefficients
Raise to a power
Multiply or divide
Combine
Exponents
Multiply
Add or subtract
Copy
Thinking of the order of operations, notice how the little baby exponents are
always one step behind the big coefficients!
The quotient to a power rule tells us that when a fraction is raised to a power,
both the numerator and the denominator get raised to the power.
2
2
4
2 2
   2 
9
3 3
1
3
2

 
2
3
Notice how an exponent of -1 gives you the reciprocal.
Scientific Notation
Scientific notation is a way of handling very large and very small numbers.
A positive number is in scientific notation if it is in the form a x 10n where a is
a number between 1 and 10 including 1 and n is an integer.
Changing from standard to scientific notation:
Numbers > 1
67,000,000.0 = 6.7 x 107
Roll the decimal so it is after the 6 and count the digits you passed.
BIG number ~ exponent on the ten is positive
Numbers < 1
0.0000023 = 2.3 x 10-6
Roll the decimal so it is after the 2 and count the digits you passed.
Don’t just count the zeros ~ you passed the 2 also!
TINY number ~ exponent on the ten is negative
Changing from scientific to standard notation:
Exponent positive
8.9 x 105
Exponent is positive so we’re making a BIG number
Roll the decimal 5 places: 890,000
Exponent negative
5.4 x 10-6
Exponent is negative so we’re making a tiny number
Roll the decimal 6 places:
0.0000054
Solving problems using scientific notation
(4,000,000)(80,000)
 4.0 x10 8.0 x10 
6
4
32.0 x1010
3.2 x101 x1010
3.2 x1011
Caution: Many people get the right answer but forget to convert it back into
scientific notation!