Elementary Algebra Exam 4 Material Exponential Expressions & Polynomials.
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Transcript of Elementary Algebra Exam 4 Material Exponential Expressions & Polynomials.
Elementary Algebra
Exam 4 Material
Exponential Expressions & Polynomials
Exponential Expression
• An exponential expression is:
where is called the base and is called the exponent
• An exponent applies only to what it is immediately adjacent to (what it touches)
• Example:
nana
23x 3 not to x,only to appliesExponent 4m negative not to m, only to appliesExponent
32x (2x) toappliesExponent
Meaning of Exponent
• The meaning of an exponent depends on the type of number it is
• An exponent that is a natural number (1, 2, 3,…) tells how many times to multiply the base by itself
• Examples: 23x 4m
32x
xx3mmmm 1
xxx 222 38xexponentinteger any of meaning learn the willsection wenext In the
Rules of Exponents
• Product Rule: When two exponential expressions with the same base are multiplied, the result is an exponential expression with the same base having an exponent equal to the sum of the two exponents
• Examples:
nmnm aaa
24 33 243 63 47 xx 47x 11x
Rules of Exponents
• Power of a Power Rule: When an exponential expression is raised to a power, the result is an exponential expression with the same base having an exponent equal to the product of the two exponents
• Examples:
mnnm aa
243 243 83
47x 47x
28x
Rules of Exponents
• Power of a Product Rule: When a product is raised to a power, the result is the product of each factor raised to the power
• Examples:
nnn baab
23x 223 x 29x
42y 442 y 416y
Rules of Exponents
• Power of a Quotient Rule: When a quotient is raised to a power, the result is the quotient of the numerator to the power and the denominator to the power
• Example:
n
nn
b
a
b
a
23
x
2
23
x 2
9
x
Rules of Exponents
• Don’t Make Up Your Own Rules
• Many people try to make these rules:
• Proof:
nnn baba
222 2323
nnn baba
!!!!TRUE! NOT
!!!!TRUE! NOT
222 2323
Using Combinations of Rules to Simplify Expression with Exponents• Examples:
43225 pm 128425 pm 128165 pm 12880 pm
3325 yx 9635 yx 96125 yx
232332 32 yxyx 6496 98 yxyx 151072 yx
252
332
3
2
yx
yx
104
96
9
8
yx
yx
y
x
9
8 2
Homework Problems
• Section: 4.1
• Page: 261
• Problems: Odd: 5 – 11, 25 – 79
• MyMathLab Section 4.1 for practice
• MyMathLab Homework Quiz 4.1 is due for a grade on the date of our next class meeting
Integer Exponents
• Thus far we have discussed the meaning of an exponent when it is a natural (counting) number: 1, 2, 3, …
• An exponent of this type tells us how many times to multiply the base by itself
• Next we will learn the meaning of zero and negative integer exponents
• Examples: 0532
Integer Exponents
• Before giving the definition of zero and negative integer exponents, consider the pattern: 1624
823 422 221
02
12
22
1
2
1
4
1
8134 2733 932 331
03 13
23
1
3
1
9
1
1
2
1
2
2
1
1
3
1
2
3
1
Definition of Integer Exponents
• The patterns on the previous slide suggest the following definitions:
• These definitions work for any base, , that is not zero:
10 an
n
aa
1
a
05 1 32
3
2
1
8
1
Quotient Rule for Exponential Expressions
• When exponential expressions with the same base are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent
Examples:
.
nmn
m
aa
a
7
4
5
5
4
12
x
x
374 55
8412 xx
“Slide Rule” for Exponential Expressions
• When both the numerator and denominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponentExample: Use rule to slide all factors to other part of the fraction:
• This rule applies to all types of exponents• Often used to make all exponents positive
sr
nm
dc
banm
sr
ba
dc
Simplify the Expression:(Show answer with positive exponents)
141
23
2
8
yy
yy
141
26
2
8
yy
yy
31
8
2
8
y
y
83
128
yy 11
16
y
Homework Problems
• Section: 4.2
• Page: 270
• Problems: Odd: 1 – 51, 57 – 77
• MyMathLab Section 4.2 for practice
• MyMathLab Homework Quiz 4.2 is due for a grade on the date of our next class meeting
Scientific Notation
• A number is written in scientific notation when it is in the form:
Examples:
• Note: When in scientific notation, a single non-zero digit precedes the decimal point
integeran is and 101 where,10 na a n
5102.3 9105342.1
201098.6
Converting from Normal Decimal Notation to Scientific Notation
• Given a decimal number:– Move the decimal to the right of the first non-zero digit
to get the “a”– Count the number of places the decimal was moved
• If it was moved to the right “n” places, use “-n” as the exponent on 10
• If it was moved to the left “n” places, use “n” as the exponent on 10
• Examples:
.
5102.3 9105342.1
201098.6
000,320 left places 5 decimal Move
3420000000015.0 right places 9 decimal Move
000,000,000,000,000,000,698 left places 20 decimal Move
na 10
Converting from Scientific Notation to Decimal Notation
• Given a number in scientific notation:– Move the decimal in “a” to the right “n” places,
if “n” is positive– Move the decimal in “a” to the left “n” places,
if “n” is negative
• Examples:
.
5102.3 9105342.1
201098.6
000,320right places 5 decimal Move
3420000000015.0left places 9 decimal Move
000,000,000,000,000,000,698right places 20 decimal Move
na 10
Applications of Scientific Notation
• Scientific notation is often used in situations where the numbers involved are extremely large or extremely small
• In doing calculations involving multiplication and/or division of numbers in scientific notation it is best to use commutative and associative properties to rearrange and regroup the factors so as to group the “a” factors and powers of 10 separately and to use rules of exponents to end up with an answer in scientific notation
• It is also common to round the answer to the least number of decimals seen in any individual number
Example of Calculations Involving Scientific Notation
• Perform the following calculations, round the answer to the appropriate number of places and in scientific notation
9
205
1053.1
1098.6102.3
9
205
10
1010
53.1
98.62.3
9205 10101059869281.14 341059869281.14
3510459869281.1 35105.1
notation? scientificin put this todo toneed wedoWhat
Homework Problems
• Section: 4.3
• Page: 278
• Problems: Odd: 1 – 9, 13 – 49, 63 – 75
• MyMathLab Section 4.3 for practice
• MyMathLab Homework Quiz 4.3 is due for a grade on the date of our next class meeting
Review of Terminology of Algebra
• Constant – A specific number
Examples of constants:
• Variable – A letter or other symbol used to represent a number whose value varies or is unknown
Examples of variables:
3 65
4
x n A
Review of Terminology of Algebra
• Expression – constants and/or variables combined with one or more math operation symbols for addition, subtraction, multiplication, division, exponents and roots in a meaningful wayExamples of expressions:
• Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables
32 x5n
104 wy 92
Review of Terminology of Algebra
• Term – an expression that involves only a single constant, a single variable, or a product (multiplication) of a constant and variables
Examples of terms:
• Note: When constants and variables are multiplied, or when two variables are multiplied, it is common to omit the multiplication symbol
Previous example is commonly written:
2 m 25 x 23 yx 5
3
2yx
2 m 25x 23 yx 5
3
2xy
Review of Terminology of Algebra
• Every term has a “coefficient”
• Coefficient – the constant factor of a term– (If no constant is seen, it is assumed to be 1)
• What is the coefficient of each of the following terms?
2
m
25x2
1
5
1 3
223 yx
5
3
2xy
Terminology of Algebra
• Every term has a “degree”
• Degree – the sum of the exponents on the variables in the term– (constant terms always have degree 0)
• What is the degree of each of the following terms?
2
m
25x0
1
2
56
23 yx
5
3
2xy
Review of Like Terms
• Recall that a term is a constant, a variable, or a product of a constant and variables
• Like Terms: terms are called “like terms” if they have exactly the same variables with exactly the same exponents, but may have different coefficients
• Example of Like Terms:
yxandyx 22 73
Review of Like Terms
• Given the term:
• Which of the following are like terms to this one?
324. xy
322 yx34xy
235 yx3
2
1xy
Adding and Subtracting Like Terms
• When “like terms” are added or subtracted, the result is a like term and its coefficient is the sum or difference of the coefficients of the other terms
• Examples:
xxx 72 x4
yxxxyyxx 2222 26194 xyyxx 6206 22
Polynomial
• Polynomial – a finite sum of terms
• Examples:
456 2 xx ?many terms How 3?first term of Degree
term?second oft Coefficien2
5-642 53 yxyx ?many terms How 2
term?second of Degree?first term oft Coefficien
103
Special Names for Certain Polynomials
Number of Terms
One term:
Two terms:
Three terms:
Special Name
monomial
binomial
trinomial456 2 xx
642 53 yxyx
yx29
Evaluating Polynomials
• To “evaluate” a polynomial is to replace variables with parentheses containing specific numbers and simplify
• Evaluate the polynomial for : 4,3 yx
224 13
224 xy
23424
984
Adding and Subtracting Polynomials
• To add or subtract polynomials horizontally:– Distribute to get rid of parentheses– Combine like terms
• Example:
233132 22 xxxxx
233132 22 xxxxx
xx 53 2
Adding and Subtracting Polynomials
• To add or subtract polynomials vertically:– Line up like terms in vertical columns– Add or subtract terms in each column
• Example: 23132 22 xxx
2 3
132 2
2
x
xx
332 xx
Homework Problems
• Section: 4.4• Page: 289• Problems: Odd: 1 – 55, 59 – 69,
73 – 77
• MyMathLab Section 4.4 for practice• MyMathLab Homework Quiz 4.4 is due for a
grade on the date of our next class meeting
Multiplying Polynomials
• To multiply polynomials: – Get rid of parentheses by multiplying every
term of the first by every term of the second using the rules of exponents
– Combine like terms
• Examples:
4523 2 xxx 12156452 223 xxxxx 12112 23 xxx
4532 xx 1215810 2 xxx 12710 2 xx
Multiplying Binomials by FOIL
• As seen by the last example, we already know how to multiply binomials by the general rule (every term of first by every term of the second)
• With binomials, this is sometimes called the FOIL method:– First times First– Outside times Outside– Inside times Inside– Last times Last
4532 xx 1215810 2 xxx 12710 2 xx
F O I L
Homework Problems
• Section: 4.5
• Page: 297
• Problems: Odd: 1 – 55, 61 – 83
• MyMathLab Section 4.5 for practice
• MyMathLab Homework Quiz 4.5 is due for a grade on the date of our next class meeting
Squaring a Binomial
• To square a binomial means to multiply it by itself
• Although a binomial can be squared by foiling it by itself, it is best to memorize a shortcut for squaring a binomial:
232x 3232 xx 9664 2 xxx 9124 2 xx
2ba 22 2 baba
232x
22 secondecond)2(first)(sfirst
9124 2 xx
Finding Higher Powers of Binomials
• To find powers of binomials higher than the second we use the definition of exponents and the rules already learned
• Example:
332x 3232 2 xx 329124 2 xxx
27183624128 223 xxxxx
2754368 23 xxx
Conjugate Binomials
• Two binomials are called “conjugates” if they are exactly the same except for the sign in the middle
• Examples: What is the conjugate of the given binomial?
:is Conjugate 32 x
:is Conjugate 45 x
32 x
45 x
Multiplying Conjugate Binomials
• Conjugate binomials can be multiplied by foil:
• However, it is best to memorize a formula for multiplying conjugate binomials:
3232 xx 9664 2 xxx 94 2 x
22 bababa lastlast and rst)(first)(fi
3232 xx 94 2 x
Homework Problems
• Section: 4.6
• Page: 303
• Problems: Odd: 3 – 19, 25 – 53
• MyMathLab Section 4.6 for practice
• MyMathLab Homework Quiz 4.6 is due for a grade on the date of our next class meeting
Dividing a Polynomial by a Monomial
• Write problem so that each term of the polynomial is individually placed over the monomial in “fraction form”
• Simplify each fraction by dividing out common factors xyxyxyyx 224128 23
xyxy
xy
xy
xy
xy
yx
2
2
2
4
2
12
2
8 23
xyyx
1264 2
Dividing a Polynomial by a Polynomial
• First write each polynomial in descending powers
• If a term of some power is missing, write that term with a zero coefficient
• Complete the problem exactly like a long division problem in basic math
Example 415032 232 xxx
40150023 223 xxxxx
150023 23 xxxx3
xxx 1203 23 402 xx
( )
xx 122 2 150
2
8 0 2 2 xx( )
15812 x
4
158122
x
x
Homework Problems
• Section: 4.7
• Page: 312
• Problems: Odd: 7 – 31, 39 – 75
• MyMathLab Section 4.7 for practice
• MyMathLab Homework Quiz 4.7 is due for a grade on the date of our next class meeting