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Exponents, Polynomials, and Scientific Notation: Rules and Operations - Prof. Zachary Hann, Exams of Algebra

A review of the rules and operations related to exponents, polynomials, and scientific notation for math 102 students. It covers the concepts of bases and exponents, exponent rules, scientific notation, monomial operations, and polynomial addition, subtraction, multiplication, and division. The document also includes examples and formulas.

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

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Test 4 Review (Ch. 5) Math 102
Ch. 5: Exponents and polynomials.
5.1-2: Exponent “rules” and scientific notation.
1. You need to know what the “base” and “exponent” are for an exponential expression; for
example, in the expression
4
2
, 2 is the base and 4 is the exponent.
2. Know all the exponent rules we developed in class and be able to apply them meticulously to
the problems in the book (it is very easy to make mistakes or invent new rules!). I highly
encourage you to work at understanding why the rules are true. Understanding the logic behind
the rules makes them easier to remember and harder to use incorrectly. Also, when the
exponents are nice enough, you don’t even have to think about the formal rules as long as you
understand how to collect factors together. I’ve written a compact list below:
r s r s
a a a
+
=
( )
r
r r
ab a b
=
(
s
r rs
a a
=
(
s
r rs
a a
=
r
r
r
a a
b b
=
r
r s
s
a
a
a
= 0
1
a
=
1
r
r
a
a
=
Note: we don’t consider an exponential expression to be “simplified” until there are no negative
exponents left in the expression.
3. Know how to put big and small numbers into scientific notation. Know how to take a number
in scientific notation and turn it back into “standard” form.
5.3: Operations with monomials.
1. A monomial is a product of constants and variables.
2. The only way you can add or subtract monomials is if the variable parts are exactly the same
(then they are “like terms”). Otherwise, you can’t add or subtract them.
3. To multiply monomials, you reorganize all the factors so that you just multiply the constants
and like variables separately. You will have to use exponent rules to simplify the variable part.
4. To divide monomials, you just cancel common factors to simplify the fraction, dealing with
constants and like variables one type at a time. Remember to express your final answer with no
negative exponents.
5. Be able to multiply and divide numbers in scientific notation and then express your answer in
scientific notation.
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Test 4 Review (Ch. 5) Math 102

Ch. 5: Exponents and polynomials.

5.1-2: Exponent “rules” and scientific notation.

1. You need to know what the “base” and “exponent” are for an exponential expression; for example, in the expression 24 , 2 is the base and 4 is the exponent. 2. Know all the exponent rules we developed in class and be able to apply them meticulously to the problems in the book (it is very easy to make mistakes or invent new rules!). I highly encourage you to work at understanding why the rules are true. Understanding the logic behind the rules makes them easier to remember and harder to use incorrectly. Also, when the exponents are nice enough, you don’t even have to think about the formal rules as long as you understand how to collect factors together. I’ve written a compact list below:

a a^ r^ s^ = a r^ +s ( ab) r^ = a br^ r ( ar )s= ars ( ar )s=ars

r (^) r r

a a b b

  (^) =  

r r s s

a a a

= − a^0 = 1 r^1 a (^) a r − (^) =

Note: we don’t consider an exponential expression to be “simplified” until there are no negative exponents left in the expression.

3. Know how to put big and small numbers into scientific notation. Know how to take a number in scientific notation and turn it back into “standard” form.

5.3: Operations with monomials.

1. A monomial is a product of constants and variables. 2. The only way you can add or subtract monomials is if the variable parts are exactly the same (then they are “like terms”). Otherwise, you can’t add or subtract them. 3. To multiply monomials, you reorganize all the factors so that you just multiply the constants and like variables separately. You will have to use exponent rules to simplify the variable part. 4. To divide monomials, you just cancel common factors to simplify the fraction, dealing with constants and like variables one type at a time. Remember to express your final answer with no negative exponents. 5. Be able to multiply and divide numbers in scientific notation and then express your answer in scientific notation.

5.4: Addition and subtraction with polynomials.

1. Addition consists of simply combining like terms from the two polynomials. 2. Subtraction requires that you distribute the factor of -1 over the second polynomial before you start combining like terms.

5.5-5.6: Multiplication of polynomials.

1. To multiply a monomial by a polynomial, you just distribute that monomial over all the terms in the polynomial. Remember to simplify each term you obtain by this method. 2. To multiply two polynomials, we add together “every possible product containing one term from each polynomial”. In the case of a product of two binomials, this is just the “FOIL” you know and love. 3. Some products are important to recognize (although you don’t really need to memorize anything):

The difference of two squares: ( x − y )( x + y ) = x 2 − y^2.

Perfect square binomials: ( x ± y )^2 = x 2 ± 2 xy +y^2

5.7-5.8: Division with monomials and polynomials.

1. To divide a polynomial by a monomial, you just divide each term in the polynomial by that monomial. Each of the fractions that results will need to be simplified using exponent rules. 2. To divide a polynomial by another polynomial, we use the long division algorithm that you learned long ago for base-10 numbers. Remember: you need to have place holders (zero coefficients) for any term that doesn’t appear in the dividend. Also, you need to know how to handle remainders – they add a fractional part to your answer.