lecture1for110 - Math 110 FS 2008 Lecture #1 CH 1.1—1.3...

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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 1.1-1.3 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 1.1-1.3 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 1.1-1.3 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—3-E—1CI 12 3 4 5 Math 110 FS 2008 Lecture #1 CH 1.1-1.3 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$4 b) —2 < x C) xS4 Math 110 FS 2008 Lecture #1 CH 1.1-4.3 Rea! number system. Factoring Polynomials. A so: ‘rhm' fiofig‘ém of ma 1%: mi: mmfim M-{Wiiflfi two goo-Em? 53:32:51 :1 at: «at: :3 .ngzmjpfio 1.3.; i5 sz-a'Elétté 2m iatermi, Epfifiiilfi notaté‘igoa“; mam .igniixmil mm rim] i3 magi to imfiimfi an iizgromi on {ho mambo? firm; Po? ommp‘fio? {319$ iHJEézf‘fEiE inflmiing 3E3: nfimmmx moi]. "E 4322 3.: 1:11“; Wiiiéfiflfl as; “‘23 The gmmfiihow so; ideE-Mn: Elam Ehré: Emon and 3 m fifi‘? imam Elf" mi? ain’t?" m to ismlimim if? 5:33;: Mai-wt :gqom 'fimxzfiwig ow? fig; ifl 31%: axiom fl'i’gifijfl? Shim sovomfi Eygjféoaé iéfiififi'fijfi, Wilma :3 it: b. 1 M Ifl'fiflfifig Emmi-on; #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 1.1-1.3 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 1.1-1.3 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 1.1-1.3 Real nnmber system. Factoring Polynomials. Lecture #1 D. Factoring the Difference of Two Squares. A2 “132 :(A+B)(A—B) Problem #4. Factor a)121x2-----4y2 b) xZ-us E. Factoring Perfect Square Trinomials. A2 +2AB+82 :(A+B)2 A2 -—2AB+BZ =(A—B)2 Problem #5. Factor a) x2-10x+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.

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lecture1for110 - Math 110 FS 2008 Lecture #1 CH 1.1—1.3...

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