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Unformatted text preview: Math 110 FS 2008 Lecture #1
CH 1.1—1.3 Real number system. Factoring Polynomials. THE REAL NUMBER SYSTEM The real number system evolved overtime by
expanding the notion of what we mean by the word
“number.” At ﬁrst, “number” meant something you
could count, like how many sheep a farmer owns.
These are called the natural numbers, or sometimes
the counting numbers. ral Numbers
or “Counting Numbers”
1,2,3,4,5,... At some point, the idea of “zero” came to be
considered as a number. If the farmer does not have
any sheep, then the number of sheep that the farmer
owns is zero. We call the set of natural numbers plus
the number zero the whole numbers. h o l e u m be re
Natural Numbers together with “zero” 0,1,2,3,4,5,... Math 110 FS 2008 Lecture #1
CH 1.11.3 Real number system. Factoring Polynomials. integers Whole numbers plus negatives
...—4,—3,—2,——1,0, 1,2, 3,4, . .. Rational Numbers All numbers of the form %, Where a and b are
integers (but I) cannot be zero). Rational numbers include What we usually call
fractions . Notice that the word “rational” contains the word “ratio, ” which Should remind you of
fractions. The bottom of the fraction is called the denominator.
Think of it as the denomination—it tells you what size fraction we are talking about: fourths, ﬁfths,
etc. The top of the fraction is called the numerator. It
tells you how many fourths, ﬁfths, or whatever. Math 110 FS 2098 Lecture #1 CH 1.11.3 Real number system. Factoring Polynomials. lrrational Numbers . Cannot be expressed as a ratio of integers.
. As decimals they never repeat or terminate. Examples: Lima
11
.C’
a]
U1 2: CI. 666 666 3. = 0.4545d535 11 ﬁ=1.41421356... rrm3.l4159265... Rational (terminates)
Rational (repeats) Rational (repeats) Irrational (never repeats
or terminates) Irrational (never repeats
or terminates) Math 110 FS 2008 Lecture #1
CH 1.1—1.3 Real number system. Factoring Polynomials. The Real Numbers . Rationais + Irrationals . . All points on the number line When we put the
irrational numbers together with the rational
numbers, we ﬁnally have the complete set of
real numbers. Any number that represents an
amount of something, such as a weight, a
volume, or the distance between two points, will
always be a real number. The following diagram
illustrates the relationships of the sets that make
up the real numbers. 2 A 11 23 Irratlnnal Natural '
1,2,3. . .
3 '
 .1 3 / 211323132345. ..
WM 3 Math 110 FS 2008 Lecture #1
CH 1.11.3 Rea! number system. Factoring Poiynomials. m: Ordered Set The real numbers have the property that they are
ordered, which means that given any two different
numbers we can always say that one is greater or
less than the other. A more formal way of saying this IS: For any two real numbers a and I), one and only one
of the following three statements is true: 1. a is less than I), (expressed as a < b)
2. a is equal to I), (expressed as a = b)
3. a is greater than I), (expressed as a > b) The um ber Line The ordered nature of the real numbers lets us
arrange them along a line (imagine that the line is
made up of an inﬁnite number of points all packed
so closely together that they form a solid line). The
points are ordered so that points to the right are
greater than points to the left: {—1—l——+—l—l——l——l—l—+——l——l——% —5—¢1—3E—1CI 12 3 4 5 Math 110 FS 2008 Lecture #1
CH 1.11.3 Real number system. Factoring Polynomials. . Every real number corresponds to a distance on
the number line, starting at the center (zero). . Negative numbers represent distances to the left
of zero, and positive numbers are distances to
the right. . The arrows on the end indicate that it keeps
going forever in both directions. A number line can be used to draw the graph of a
set of numbers. Graph all real numbers at such as
@«Q<x$4
b) —2 < x C) xS4 Math 110 FS 2008 Lecture #1
CH 1.14.3 Rea! number system. Factoring Polynomials. A so: ‘rhm' ﬁoﬁg‘ém of ma 1%: mi: mmﬁm M{Wiiﬂﬁ two gooEm? 53:32:51 :1 at: «at: :3 .ngzmjpﬁo 1.3.; i5 sza'Elétté 2m iatermi, Epﬁﬁiilﬁ notaté‘igoa“; mam .igniixmil
mm rim] i3 magi to imﬁimﬁ an iizgromi on {ho mambo? ﬁrm; Po? ommp‘ﬁo? {319$ iHJEézf‘fEiE
inﬂmiing 3E3: nﬁmmmx moi]. "E 4322 3.: 1:11“; Wiiiéﬁﬂﬂ as; “‘23 The gmmﬁihow
so; ideEMn: Elam Ehré: Emon and 3 m ﬁﬁ‘? imam Elf" mi? ain’t?" m to ismlimim if? 5:33;: Maiwt :gqom 'ﬁmxzﬁwig ow? ﬁg; ifl 31%: axiom ﬂ'i’giﬁjfl?
Shim sovomﬁ Eygjféoaé iéﬁiﬁﬁ'ﬁjﬁ, Wilma :3 it: b. 1
M Iﬂ'ﬁﬂﬁﬁg Emmion; #5 mm mm {is dﬁmﬁm 3m maria as m 3:3? of 31% imm Wifﬁi’é. .1;  Eﬁmmi i3 Wﬁiﬁiﬁﬂ 5E§Em m 313. m5 mama
{w Mme .ﬂﬂiiﬁiﬂiﬂr era; WﬁiEﬁ’ﬂ Math 110 FS 2008 Lecture #1
CH 1.14.3 Real number system. Factoring Polynomials. Absolute Value The absolute value of a number is the distance from
the point that corresponds with the number to the
origin (zero) on the number line. That distance is
always given as a non—negative number. Formal definition. The distance d between two points a and b on the
number line is d z {a — bl. Find the distance between the points on the number line that correspond to numbers
a) 10 and2 b)7and —3. 0) Graph the interval which corresponds with
{x:x]>4}; {lexlsél}; {x223lxl<4} Math 110 FS 2008 Lecture #1
CH 1.11.3 Real number system. Factoring Polynomials. Factoring Polynomials. I Deﬁnition of a Polynomial in x. A polynomial in x is an algebraic expression in the
form 11 11—1 11—2
anx + aﬂx + anhzx +    + (1195 + Clo, Where an, a a , a1, and do are real numbers, 11—19 11—27 an 72 0,
n is a nonnegative integer.
n is the degree of polynomial
an is the leading coefﬁcient, are is the constant term. Math 110 FS 2008 Lecture #1
CH 1.11.3 Rea! number system. Factoring Polynomials. 1" Factoring is the process of writing a polynomial as
the product of two or more polynomials. We will
do factoring with integer coeﬂicients.
Polynomials that cannot be factored using integer
coefﬁcients are called irreducible over the
integers, 0r prime. * Methods of Factoring. A. Factoring out the Greatest Common Factor.
Problem #1. Factor a) 25x5 ~15x3 b) 3x(x—2)—24(x—2) B. Factoring by grouping.
Problem #2. Factor x4 5x3 *3x+15
C. Factoring Trinomials ax2 + bx + (2. Problem #3. Factor a) x2+5x+6 b) 6x2+13x—5 lO Math 110 FS 2008
CH 1.11.3 Real nnmber system. Factoring Polynomials. Lecture #1 D. Factoring the Difference of Two Squares.
A2 “132 :(A+B)(A—B) Problem #4. Factor a)121x24y2 b) xZus E. Factoring Perfect Square Trinomials.
A2 +2AB+82 :(A+B)2
A2 —2AB+BZ =(A—B)2
Problem #5. Factor a) x210x+25 b) 2x3+12x+18 F. Factoring Sums and Differences of Two Cubes. A3+B3=(A+B)(A2—AB+BZ) A3 —B3 =(A—B)(A2 +AB+Bz) 11 ...
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This note was uploaded on 03/15/2009 for the course MATH 110 taught by Professor Wormer during the Spring '08 term at Michigan State University.
 Spring '08
 WORMER

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