Addition and Subtraction of real numbers (1.3 & 1.4)

Transcription

1 Math 051 lecture notes Professor Jason Samuels Addition and Subtraction of real numbers (1.3 & 1.4) ex) = ex) = ex) 12-4 = ex) 7-9 = ex) -3-4 = ex) 6 - (-2) = ** CHECK YOUR SIGNS ** IDENTIFY YOUR OPERATION

2 ** ex) -5 - (-3) = ex) 7 + (-2) = you do: ex) 12 - (-7) = ex) = ex) = ex) = ex) [(7-5) - (11-6)] = Multiplication and Division (1.6 & 1.7)

3 ex) 3 2 = ex) (4)(-3) = ex) (-3)(-6) = you do: ex) (6)(-5) = whats the operation? operation: multiplication (also division) sign: negative times negative is positive negative times positive is negative positive times negative is negative Compare: ex) 6-3 = ex) 6(-3) = you do: ex) -7-4 = ex) (-7)(-4) = some division: ex) 12 4 = ex) 21 (-7) = ex) 20 = -4 ex) -16 = -8

4 parentheses: might mean - do me first - multiply - sometimes both - sometimes neither order of operations (or, do them all together) ex) = ex) (4)(-5) + (-3)(-6) = ex) (3+6)4-8 you do: ex) 3-2(6-8) = Exponents ex) (2)(2)(2)(2) can also be written as: ex) 3 6 is the same as: ex) 5 0 = ex) 1 can also be written as: (6)(6)

5 opposite, reciprocal, absolute value (1.2) adding what number doesnt change what you have? 0 ("additive identity") ex) what do you add to -3 to get 0? thats called the additive inverse, or "opposite" multiplying by what number doesnt change what you have? 1 ("multiplicative identity") ex) what do you multiply by 5 in order to get 1? 1 / 5 thats called the multiplicative inverse, or "reciprocal" ex) whats the opposite of 7 / 5? whats the reciprocal? ex) what is 3? the absolute value of 3 make it a positive number (or zero) ex) -3 = ex) 2-5 = ex) 0 = 0 Properties of real numbers (1.5) commutative property ex) for addition: = ex) for multiplication: (5)(7) = (7)(5) associative property ex) for addition: (9+5)+6 = 9+(5+6) ex) for multiplication: (3 2) 7 = 3 (2 7)

6 distributive property ex) 3(4+11) = 3(4) + 3(11) ex) -4(7-3) = do it again... with fractions (1.9) Q. what is a fraction? -part of a whole -means of division -ratio -decimal -percent -rational number -one integer "over" another integer Add/Subtract fractions Ex) + = Ex) + =

7 You do: + why does adding fractions work like that? - here's the picture Ex) + + = Need same denominator. Note that there is a 3 in the 6, so just find the LCD for 7,6 its 42. Multiplication & division with fractions calculate & simplify: ex) ex)

8 ex) 10 notice that 10 2 = 5 so dividing by a number is the same as multiplying by the reciprocal ex) 6 ex) why does fraction multiplication work like that? - here's the picture Mixed fractions Calculate and simplify: Ex) Calculate and simplify:

9 Ex) ( ) Ex) ( )( )

10 why do we use variables? - to represent (unknown) numbers - to help solve an equation - to show a general relationship ex) Sam is 18, Jason is 33 so, when Sam is 40, Jason is when Sam is x, Jason is two very DIFFERENT situations: expressions & equations we know expressions from arithmetic: ex) 4(7-5) what do you do? you calculate note that your final result is a number - feels like an answer there are expressions in algebra: ex) 3(2x-7) + 4 what do you do? you simplify. [why?] note that your final result has x in it - not very satisfying heres an equation: x + 5 = 8 how are EXPRESSIONS and EQUATIONS different? suppose you have 3x = 12 what do you do? you want to SOLVE FOR x you get x=4 you cant do that with expressions ex) 5x+20 what do you do?

11 nothing (unless someone tells you to do something) 3x=12... what operation did we do? divide both sides by 3 can you do something like that with 5x+20? no: 5x+20 cannot magically become x+4 if you have an expression, there is no "=", and it cant magically appear Evaluating expressions evaluate the following expressions ex) x 2 +4x if x=5 ex) 7 - x 3 + 3x if x= -2 ex) 2(4x-5)+3 if x=4 ex) 1 / 2 gt 2 if g=10, t=3 good general rule: use parentheses when you substitute (plug in) a value

12 ex) 4(x-3) 2 if x=5 ex) x / 4 + 4x if x=-8 a word about exponents, parentheses, and signs ex) (-3)(-3)= ex) -3 3 = rewrite them with exponents: (-3) so these are NOT THE SAME on the left, you square everything, including the negative on the right, you only square the 3 ex) (-2) 4 = ex) (-x) 4 = lets talk about expressions (2.1) ex) 2x x "pieces" are called terms they are separated by addition and subtraction compare: ex) 3 + x - 2 ex) 3x-2 what are "like terms"?...terms you can combine simplify:

13 ex) 5+2x ex) 2x+3x ex) 3+2(4-3x) parentheses are in the way - so distribute ex) 4x+3[2(9-3x)+4] now, ex) can you simplify x 3 + 3x - 2? are there any like terms? ex) 5m 2 + 3m - 2m 2 + m ex) 3x 2 + 2x 2 ex) 6t t 2 + t + 2 Solving linear equations (2.4, includes 2.2, 2.3) what does it mean to be "linear"?...if the expression only has x or y, nothing like x 2 or xy 2 (can be any variables, of course)

14 solve for the variable: ex) x + 2 = 7 ex) x + 24 = 47 ex) t - 8 = 27 ex) x- (-4) = 15 ex) 4 = x - 19 ex) 2x = 6 ex) 12x = 48 ex) x = 15 3 ex) 7x = 84

15 ex) x = more solving for the variable ex) 2x + 7 = 15 you want x by itself - what do you get rid of first? The 7, because we need to undo that side by order Of operations ex) 3x - 5 = 5x + 9 cannot get x by itself directly until we remove x from one side ex) 4(x+2) = 2(3x+1) want to get x by itself, but the parentheses are in the way distribute to get rid of the parentheses

16 you do: ex) -4x-3 = 13 ex) x+7 = 7x+31 ex) 2(4-x)+4 = 3(2x-4) now, with fractions ex) + 3 = 11 ex) = 24 you do: ex) = 55 solving for a variable, when there are two variables (2.5) if you are given the value for one variable ex) 2x - 5y = 30 solve for y if x=3

17 (thats just like the other problems) ex) 3x + 2y = 12 solve for x actually, this works exactly the same to get x by itself, what do you need to get rid of? ex) x = 6 solve for x y ex) s = 4 t solve for t ex) y+3 = 4 solve for y x

18 Solving word problems (2.6, 2.7) you have to TRANSLATE between words and math ex) you have three. then you get two more. how many do you have now? ex) Dave has a certain amount. he gets seven more. how many does he have now? ex) Alan has the same amount. it triples. how much does he have now? ex) Tanya has the same amount. it doubles. then she gets four more. how much does she have now? ex) How much do Dave, Alan and Tanya have together? percents ex) 50% of 12 is what? ex) 15% of 80 is what?

19 ex) 20% of what is 15? ex) at a restaurant, you got great service and want to give a 20% tip on a $60 bill. how much is the tip? ex) at the pharmacy you buy some aspirin. the sales tax on the purchase was $2. sales tax rate is 8%. how much was the aspirin sticker price? you can do the calculations using decimals or fractions by hand, i think fractions are easier by calculator, decimals are easier...its your choice

20 ex) a rectangle has a length 5 inches longer than the width. the perimeter is 58 inches. what are the dimensions of the rectangle? ex) Joel has twice as many nickels as dimes, all together worth a total of $1.80 how many does he have of each? Advice: first, give each unknown a name next, write down the relationship ex) two consecutive integers add to 45. what are the numbers?

21 ex) five consecutive integers add to 85. What are the numbers? you do: joseph has one more than twice as many toys as kara. together they have 46 toys. how many toys does each have? inequalities (2.8, 2.9) ex) solve for x: 2x = 6 graph the solution on the number line: now, ex) solve for x: 2x > 6 solve it exactly the same way

22 graph the solution on the number line: ex) 3x-7 < 1 everything is the same, with ONE EXCEPTION which is bigger? ex) 3 4 multiply both sides by (-2) ex) -6-8 so, when you multiply (or divide) by a negative number, the inequality switches note that this is the only time ex) 3<4, subtract 1 from both sides, you still have 2<3 ex) -3x < 15 you do: ex) solve and graph: -2x + 5 < 17 compound inequalities [optional] how do we write "x is between 2 and 5"? thats the same as saying that its bigger than 2, and also less than 5 we could write: x>2 and x<5 here is a shorter way: take this 2 < x combine it with this x < 5

23 to get this 2 < x < 5 solve: ex) 2 < 3x-7 < 14 its easy, do them both at the same time

24 linear equations, representing and graphing (3.1-5) ex) cell phone monthly bill Shyanna's bill: $40 (and all the calls she wants) Tarik's bill: 10cents/min Jerry's bill: $30 plus 5cents/min which plan is best?...depends how much you talk what about 250min? Shyanna: $40 note: in terms of dollars "10 cents" is ".10" Tarik: (250)(.10) = $25 Jerry: (.05)(250) = = $42.50 what about 550 min? Shyanna: $40 Tarik: (.10)(550) = $55 Jerry: (.05)(550) = = $57.50 how can we represent this information? - numerically # minutes Shyanna Tarik Jerry graphically Shyanna Tarik Jerry

25 how much does it cost to talk 0 minutes for: Shyanna? Tarik? Jerry? what does that have to do with the graph?...where x=0 is where the graph crosses the y-axis - that is called the y-intercept Tarik pays.10 dollars per min, and he also pays $10 for 100 minutes is that the same thing? how can we check? in fact, we can do this with any two points and this tells us the rate, or slope ex) for Tarik: 250 minutes cost $ minutes cost $45 ex) what is the rate for Jerry? 350 minutes cost $ minutes cost $52.50 ex) what is the rate for Shyanna? 250 minutes cost $ minutes cost $40 note that this is different from calculating y / x...thats assumes that we started at (0,0), which may not be true [economics sometimes uses this calculation] for Shyanna's plan, what is her rate? is it always the same? for Tarik's plan, what is his rate? is it always the same? for Jerry's plan, what is his rate? is it always the same?

26 a linear function has the same rate (or slope) everywhere what is the function for each plan? x=minutes, y=$ Shyanna: Tarik: Jerry: in general, for a linear function, we can write y = mx + b m = slope (or rate) b = y-intercept examples of linear functions ex) y = 3x+2 graph: slope= y-intercept= some solutions: x y ex) y = 2x-1 slope = y-intercept =

27 ex) y = -2x+3 slope = y-intercept = Suppose you have two points, call them (x 1 y 1 ) and (x 2 y 2 ) slope = change in y = y 2 - y 1 change in x x 2 - x 1 also called rise/run ex) what is the slope between (2,3) and (5, 10)? if slope is positive: ~ line is increasing (going up from left to right) if slope is negative, ~ line is decreasing (going down) ~ the graph will go "down & right" (not down & left) if slope is 0, graph is flat (e.g. Shyanna s plan) also equations of a line: ex) 2x + 3y = 12 graph: slope: y-intercept: solutions:

28 ex) x - 3y = 6 graph: slope: y-intercept: solutions: ex) find the slope from (1, 3) to (4,-1) ex) find the slope at (4, 2) impossible: need change finding the equation of a line (3.6) find the equation of a line given the slope and y-intercept ex) slope=3, y-intercept=4 (pretty easy) find the equation of a line given the slope and any point ex) slope=2, point is (3,4) method 1: use y=mx+b we know the slope, so we have y = 2x+b we can plug in the values from the point and it will satisfy the equation

29 4 = 2(3) + b 4 = 6 + b -2 = b so the equation is: y = 2x - 2 method 2: the answer is y - 4 = 2(x-3) first, where does that come from? second. how can there be two different answers? second question first: its really the same formula: y-4 = 2(x-3) y-4 = 2x-6 y = 2x-2...aha! now, where did that come from?...it comes from the slope (here's the explanation, if you are curious) we know that the slope is 2 2 = y 2 - y 1 x 2 - x 1 but we have values for one point, so (x 1 y 1 ) = (3,4) 2 = y 2-4 x 2-3 also, we want this to be true for any point on the line, that is (x,y) 2 = y - 4 x - 3 cross-multiply to get y - 4 = 2(x-3) or, just remember the formula: y - y 1 = m(x - x 1 ) ex) find the equation of the line with slope= -4, through the point (5,2) find the equation of the line using two points ex) find the equation of the line through (2,7) and (4,13) we can find the slope:

30 now we have the slope and a point (pick either one), so its like the problem we just solved you do: ex) find the equation of the line through (-2,3) and (1,9) special cases -horizontal lines graph: whats the slope? whats the equation? -vertical lines graph:

31 whats the slope? whats the equation? note that the slope is undefined - you cannot write the equation of this line as y=mx+b you do: ex) Find the equation of the line through (-2,5) and (1,-1), and graph it. What is the slope?

32 Solving TWO linear equations with TWO variables three ways: by graphing (4.1), by elimination (4.2), by substitution (4.3) here's what a problem might look like: ex) suppose you sell concessions at a cinema. you sell small sodas for $2 and large for $3. you ran out of small cups, so you only have one size cup today (for a small order, you fill it halfway). at the end of the day, you sold 86 sodas for $191. your manager wants to know how many of each size you sold. can you tell him? what do we know? the situation: - two variables - two equations...what do you do? simplify the situation so you have one variable and one equation (...once you have that, you know what to do)

33 was that magic? no, lets learn how to do it. ex) x = 6 y = 2x+3 solve for x & y ex) y = x-2 y = 2x-6 (thats too easy) solve with algebra: solve with a graph: the solution tells us the values for x,y that work in both equations ex) x+y=3 & x-y=1 solve by graphing

34 ex) 2x+y = 7 y = 4x-11 solve for x and y how do we solve this? ex) x - 2y = -7 solve for x & y 3x + 2y = 3 You do: Ex) solve for x & y 3x-y=14 x+3y=-2

35 do it again, a quicker way ex) x - 2y = -7 3x + 2y = 3 thats too easy - are you allowed to do that? consider: a = 3 b = 2 a+b =? c - d = 7 d = 2 c =? you do: ex) 2x - 3y = 5 x + 3y = 7 note: when you solve for the second variable, you need to plug a value into an equation. you have two choices, it doesnt matter which equation you use now, ex) 2x - y = 9 x + 2y = 2 what do you need to have so that one variable gets eliminated?

36 ex) 2x + 5y = 9 4x + 5y = 9 Note that it doesn t matter which choice you make, to eliminate x or eliminate y, you wind up with the same answer in the end. you do: ex) 2x + 3y = 5 3x + y = 11 note: some tricky person might put x first in one equation and y first in the other......make sure x's are lined up and y's are lined up when is it easier to you use... substitution?...when one variable is already by itself

37 elimination?...when one variable is set up to cancel (the coefficients are same number, opposite sign) ex) 2x - 4y = -10 3x + 2y = 1 solve for x & y special cases: ex) x+2y=3 x+2y=4 ex) 2x-3y=4 4x-6y=8 ~~~ same thing, with word problems (4.4)

38 ex) John has $1.70 in dimes and nickels with 22 total coins. how many of each does he have? ex) at a cinema, adult tickets are $12, child tickets are $7. if 110 tickets are sold for $1150, how many adults and children came? ex) the difference between two numbers is 12. their sum is 38. what are the numbers?

39 ex) one number is one more than triple another number. their sum is 29. what are the numbers? ex) two consecutive integers add to 45. what are the numbers? ex) two consecutive even integers add to 74. what are the numbers?

40 Ch5 Polynomials 5.1 exponents Ex) Arithmetic: (3)(3)(3)(3) = 3 4 Ex) algebra: (x)(x)(x) = x 3 Rules of exponents Ex) =? = 2 7 Ex) =? 5 11 Algebra: Ex) x 2 x 3 =? = x 5 Rule: x a x b = x a+b when you multiply (with the same base), add the exponents Ex) c 5 c 4 = What about: Ex) =? = There is nothing to combine Ex) =? Ex) =? Rule: = Ex) =? Exponent =1 is the same as no exponent. Simplify:

41 Ex) You do: Ex) Simplifying: Ex) x 3 + x 3 = Ex) 5x 2 + 3x 2 = you can add together like terms, which have the same base and exponent Note: you CAN combine this: you CANNOT combine this: x 3 x 4 2x 4 +3x 4 x 3 + x 4 Negative exponents Ex) simplify Negative exponent means 1 over or reciprocal OR flip between numerator and denominator Ex) 3 2 = Ex) 3-2 = Ex) = or x -4 Simplify: Ex) Ex) Ex)

42 New rule: Ex) (3 2 ) 3 = (3 3) 3 = (3 3)(3 3)(3 3) = 3 6 Where did the 6 come from?... from three two s, so 3 2 Rule: (x m ) n = x m n when you have something to a power to a power, multiply the powers Simplify: Ex) (b 3 ) 4 = Ex) (a 4 ) 5 = Ex) (x 3 ) -2 = Now, Ex) (x 3 y 4 ) 2 = (x 3 y 4 )(x 3 y 4 ) = x 6 y 8 same as above, and you distribute the exponent (exponent distributes over multiplication) Ex) (a 2 c 7 ) 3 Ex) Ex) Ex) Ex) Note: what you CAN do what you CANNOT do (x 3 y 4 ) 2 = x 6 y 8 (x 3 + y 4 ) 2 x 6 + y 8 you must FOIL An exponent distributes over multiplication (and division), NOT addition (and subtraction) 5.3 polynomials introduction- terms How do you add x 3 +3x 2 +5 and 2x 3 +4x+6? (x 3 +3x 2 +5) + (2x 3 +4x+6) How do you simplify the result? Some things can be combined. We will learn what those things are.

43 What is a term? 2x 3 +4x+6 has three terms: 2x 3 and 4x and 6 Terms start and stop at addition (or subtraction) If the terms only include whole number powers for the variables, then the whole expression is called a polynomial Ex) how many terms in the polynomial x 4-3x 5-2x+4? 4 terms: x 4, 3x 5, 2x, 4 Ex) how many terms in the polynomial x 3 x 2? One term, since it is joined by multiplication Note that many people talk about combining terms. When you do so, make sure you are specific about the operation. Different operations combine in different ways. For example, what does it mean to combine 3x 2 and 2x 2? It depends on the operation Ex) 3x 2 + 2x 2 = 5x 2 Why?...because if x 2 is an apple, you start with 3 apples then add 2 more apples, so you have 5 apples Ex) 3x 2 2x 2 = 6x 4 Why?...because 3x 2 2x 2 = 3 2 x 2 x 2 = 6x 4 Can you combine x 2 and x 3? naturally, it depends on the operation Ex) x 2 + x 3 cannot be combined Why?...because if x 2 is an apple and x 3 is an orange, when you add them together, you have 1 apple and 1 orange Ex) x 2 x 3 = x 5 Why?...because x 2 x 3 = x 2+3 = x 5 Terms which can be added together are called like terms. To be like terms, each variable and its exponent must match. Combine and simplify Ex) 3x 2 +5x 2 Ex) x 3 7x 3 Ex) x 3 + 2x x 3 + 2x

44 Ex) 4r 3 4r r 2 + 2r 3r 3 Ex) 3c 2 + 5c 2c 2 6 9c 5.4 polynomial addition & subtraction Lets do some calculations. Addition: Ex) add x 3 + 3x with 2x 3 + 4x + 6 Method A: match like terms with lines Method B: stack them A: B: Ex) x 4 3x 3 + x 6 added to 6x 3 4x 2 6x + 3 \ Compare this with stacking for arithmetic: Note that 3 and 1 get added, 2 and 4 get added. Subtraction: Ex) start with x 3 + 4x 2 + 7x + 9 and subtract 2x 2 + 5x + 4

45 Ex) start with 2x 3 +5x 2 +2x+8 and subtract 3x 3-2x 2 +4x-6 One technique for subtracting a polynomial is to flip the signs and then add. Ex) (x 2-3x+2) (3x 2 +5x-3) Ex) subtract x 2-4x+7 from 3x 2-2x-6 Note that subtract A from B means B A, so watch out for those words. 5.5 polynomial multiplication Calculate and simplify Ex) 3x 2 4x 3 Ex) 3(x+4) Ex) 2x(x 2-5) Now, lets go to the next level. Ex) (x+2)(x+3) What gets multiplied by what? Method A: FOIL Method B: stack them FOIL: First, Outer, Inner, Last

46 (x+2)(x+3) = Stacking: x+2 x+3 3x+6 x 2 +2x. x 2 + 5x + 6 Compare with stacking for arithmetic: 21 X Calculate and simplify: Ex) (2x+3)(x-4) In the previous examples, each polynomial had two terms. What about if one had more terms? Ex) x 2-4x+3 times 2x+4 By FOIL : By stacking:

47 You can use whichever method you like. 5.6 polynomial multiplication, special cases Ex) (x+4) 2 Again: exponents distribute *over multiplication*, not over addition. Ex) (r-5) 2 Ex) (3x+2) 2 Ex) (x+6)(x-6) 5.7 polynomial division (by one term) Recall from arithmetic that, for example, + =. Here, we will do the same thing with algebra, but going backwards. Calculate and simplify Ex)

48 Ex) Ex) Ex) Ex) Ex) divide 12x 5 6x 4 + 4x 2 by 2x 2

49 CH6 Factoring To understand the use of the next topic, try to solve the following equation: 35 + x 2 = 12x Two terms involve an x, they are not like terms so you cannot combine them. Therefore, it is impossible to get x by itself unless we learn a new technique. That technique is factoring. What is factoring? it s the opposite of multiplying out Recall: ex) multiply 3x(x+4) answer: 3x x ex) multiply (x+5)(x-7) answer: x 2-2x - 35 now: ex) given 3x x, we need to get 3x(x+4) ex) given x 2-2x - 35, we need to get (x+5)(x-7) how do we do that? 6.1 factoring - find the gcf factor: ex) x 2 + 3x ex) 3x 4 + 7x 3-4x 2 ex) 4p p + 20 ex) 6r 3-18r 2 + 6r ex) 6a 2 b 3 + 2ab 4 + 2a 4 b 2 look at: coefficients

50 "a" exponents "b" exponents - use the smallest exponent you do: factor ex) 8x 3-16x x 2 ex) 12r 3 s 4-6rs some basic factoring x 2 + 5x + 6 = (?? ) (?? ) how do we do this? first we make an observation: the lead term is x 2... (what) times (what) will give x 2 x 2 + 5x + 6 = (x )(x ) how do you do this?...first, look at last coefficient, figure what two numbers multiply to that (many possiblities) next, look at middle coefficient, find which pair add to that ex) x 2 + 6x + 8 ex) x 2 - x - 12 ex) x 2 + 2x - 8 ex) x 2-49

51 ex) x x + 25 note: sometimes you CANNOT factor ex) x 2 + 2x + 8 try to factor it: ex) x 2 + 4x and when i say "CANNOT", i mean you dont know how. you can factor this with advanced techniques which are beyond the scope of this course ex) factor 6r 3-18r r you do... factor: ex) 2x x + 40 ex) 4x 2 + 4x - 24

52 HOW TO FACTOR (so far) 1. find gcf (greatest common factor) 2. lead term is x 2 write factors as "x and x" 3. product: break last term down into factor pairs 4. sum: look for the pair which adds/subtracts to the middle term 5. check 6.3 factoring - when the lead coefficient is NOT 1...a little harder ex) 6x x factor out a gcf, if there is one 2. multiply first and last coefficient 3. factor that into number pairs, look for which ones add to the middle number 4. rewrite the original polynomial by breaking up the middle term 5. for the first two terms and the last two terms, separately factor the gcf 6. factor again (if you did it right, the same factor will appear twice) 7. check

53 ex) factor 4x x + 15 how could this question be asked on an exam? "factor completely" "which of the following is a factor of 4x x + 15? a) 5x+2 b) 2x+2 c) 2x+3 d) 2x+1 e) 4x+5 you do: ex) factor 10x 2 - x - 3 Note: dont forget the +1. Always check your work by multiplying. Note: you might have to factor out a negative in step 4->5 in order for the terms to match for step 5->6

54 You do: Ex) factor 3x x + 8 Ex) factor 2r 2 r - 15 recall: ex) factor x 2-81 now, ex) factor 4x 2-25 ex) factor 9r 2-16 Use this one-step shortcut when the middle term is missing. what about: x (x+6)(x+6) = x 2 +6x+6x+36 = x2 + 12x + 36 X (x+6)(x-6) = x 2 +6x-6x-36 = x2 36 X

55 Cannot factor! *Check your signs* Ex) factor 6x 2 18x + 12 Ex) factor 12a 4 b 2 6a 2 b 2 + 9a 3 b 4 Ex) factor 5x 2 45 Ex) factor 4x 2 + 3x using factoring to solve equations ex) solve for x: (x-2)(x+3) = 0

56 what happens if you try to get x by itself? As we saw before, we will get an x term and an x 2 term, and we cannot get x by itself. We need a new method... solve for x: (x-2)(x+3) = 0 something times something equals zero...so one of them must be zero - aha! x-2=0 or x+3=0 x=2 or x=-3 you get two solutions! how can we use that technique in other sorts of problems? ex) solve for x: x 2 + x - 6 = 0 moral of the story: suppose you have an equation where you must solve for x if the equation is NOT LINEAR (for example, if there is x 2 ), you cannot get x by itself if it is a polynomial, the way to solve this is to FACTOR notice that this problem 2x + 3 = 5x is different from 2x = 5x...to do the second one you need to factor (that little 2 makes a big difference) solve for x: ex) x 2 - x - 12 = 0

57 ex) 2x 2 - x - 10 = 0 ex) x 2 + 2x - 7 = 1 ex) x 3-5x 2 + 4x = 0

58 challenge exercise: make an equation with 5 solutions challenge: make an equation with solutions x= 2,5,-3 a review of factoring: ex) factor x 2 + 2x - 24 ex) factor 6x x - 10 solve for the variable: ex) s 2 + 2s = -1 ex) 8x 3-18x 2-18x = 0

59 Factoring expressions with more than one variable We will now see how to factor more complicated expressions with two variables. Lets look at two cases, two terms (difference of squares) and four terms (factor by grouping). Recall ex) factor 4x 2 81 = (2x+9)(2x-9) This special case works because (2x) 2 = 4x 2 and (9) 2 = 81 Now, Ex) factor 4x 2 81y 2 We can use the same approach. Since (2x) 2 = 4x 2 and (9y) 2 = 81y 2, this will factor as: (2x+9y)(2x-9y) Check: (2x+9y)(2x-9y) = (2x)(2x) + (2x)(-9y) + (9y)(2x) + (9y)(-9y) = 4x 2 18xy + 18xy 81y 2 = 4x 2 81y 2 Ex) factor 16x 2 49y 2 Ex) factor 45x 3 125xy 2 Here are some examples with 4 terms, which are solved with factoring by grouping into pairs. Ex) xy + 3x + 4y + 12 Ex) 6wy - 4wz + 15xy 10xz Ex) 2ac 5ad + 6bc 15bd

60 ch7 rational expressions 7.1 reducing Reduce and simplify Ex) = = Ex) = Ex) We call it cancelling, but really we are dividing. You CAN reduce by dividing: you CANNOT reduce by dividing: terms: things we are adding factors: things we are multiplying Since factors are joined by multiplication, you can cancel by dividing Since terms are joined by addition, you cannot cancel by dividing Now We know we can simplify: Replace 5 with 4+1: ( ) ( ) We can still simplify! Even though 4+1 has addition in it, (4+1) is a factor because it is connected to 2 and 7 by multiplication. Ex) simplify ( ) ( )

61 Ex) simplify ( )( ) ( )( ) Note: you cannot reduce x & x why? You cannot reduce 8 & 4 why? Ex) simplify ( )( ) ( )( ) Now, Ex) simplify Ex) simplify One tricky case: Ex) simplify Ex) simplify ( ) Ex) simplify Ex) simplify ( ) Ex) simplify

62 Ex) simplify if there are *signs* and you are not sure, multiply one factor by (-1) and see if it gives you the other factor ex) simplify You can check your answer by plugging in any number, make sure the value from the original expression agrees with the value from your answer. Check: = + 6? Ex) simplify, then check by plugging in a number:

63 Ch8 Roots and Radicals What is a root? the opposite of a power 3 2 = 9 the second power of 3 is 9 9=3 the second root (square root) of 9 is 3 Calculate and justify: Ex) 16 = because = Ex) 49 = because = Ex) = because = Ex) = because = Another kind of root 2 3 = 8 the third power of 2 is 8 8 = 2 the third root (cube root) of 8 is 2 Calculate and justify: Ex) 125 = because = Ex) 64 = because = Ex) = because = Ex) = because = Fractional Exponents represent powers and roots = x 6/3 = x 2

64 x 6/3 ex) 8 2/3 = 8 = 64 = 4 Ex) 9 1/2 = 9 (we don t actually write it like that) = 9 = 3 Note that a square root is the same as an exponent of ½ this will be useful Ex) 4 3/2 = 4 = 64 = 8 OR ( 4) = 2 3 = 8 Note that you can calculate the root then the power, or the power then the root. On an exam, if you see a cube root, you can assume that you only need to know one of these simple ones: 1 3 = = = = = 125 Roots and signs Ex) 4 2 = 16 Ex) (-4) 2 = 16 Ex) 16 = 4 because means take the positive square root What is 9? 3? 3 2 = 9 (-3)? (-3) 2 = 9 This is not a real number. It is a complex number, which is beyond the scope of this course. It is different for cube roots of negative numbers: 2 3 = 8 8 = 2 (-2) 3 = -8 8 = -2

65 Simplify: Ex) 81 Ex) 100 1/2 Ex) Ex) 16 3/2 Ex) 27 Ex) 36 Combine and simplify roots & radicals, including multiplication and division How can we simplify: 16 Remember that roots are like exponents, so it s the same as (16x 4 ) 1/2 And we know that exponents distribute over multiplication 16 1/2 (x 4 ) 1/2 Now work on it piece by piece 16 1/2 = 16 = 4 (x 4 ) 1/2 = x 4 ½ = x 2 So the answer is: 4x 2 Now we know roots work like exponents so they distribute over multiplication and we can write the steps quicker: Ex) simplify 100 = 100 = 10x 4/2 y 6/2 = 10x 2 y 3 Ex) Simplify 8 = 8 = 2x 12/3 = 2x 4 Ex) =

66 = = / / Ex) Another technique to simplify roots Ex) 5 20 We cannot simplify the roots as written, but we can if we combine them. = 5 20 = 100 = 10 Now, simplify in the same way, but with algebra. Ex) 12 3 = 36 = 36 = 6x 2 Ex) = = 5 16 = 5x 4x 4 = 20x 5 What can you do when the radical does NOT simplify nicely? Ex) 45 = 9 5 = 9 5 = 3 5 How could we know we should do that? Square roots of perfect squares simplify nicely, so we have to find the perfect square hiding inside the radical. We find it by factoring. Ex) 80

67 Now, do it with algebra Ex) = = = x 3 Note: from the first step, you could also just write it as x 7/2. But we will need this technique in the next section. Ex) 5 50 Ex) 8 48 Ex) 3 16 Simplifying roots, adding and subtracting Recall that this is NOT the same as this So how can we simplify radicals? Ex) Ex)

68 Ex) Ex) Ex) Recall that you cannot combine here Now, Ex) At first it looks like we cannot combine by adding because they are not like terms. BUT. We can manipulate the radical and then see what happens. 12 = 4 3 = 4 3 = 2 3 Now, = = 6 3 The key is to factor out the part with a nice square root. Ex) = = = = 7 2 Ex) Now, with algebra Ex) Ex)

69 Ex) Cannot combine not like terms since the roots don t match Note that for multiplication, you do not need like terms to combine, but for addition, you do need like terms. Same idea here, its just that the terms are more complicated (because of the radicals) ex) ex) ex) Ex)

70 Scientific Notation Sometimes we need to write the number for something very large, like the mass of the sun, or something very small, like the width of an atom. This often happens in science. Mass of the sun = kg Width of an atom = meters Trying to read that number with all those zeroes can make you cross-eyed. Scientists came up with a notation to make those numbers easier to read. Mass of the sun = x kg Width of an atom = 2.0 x meters The exponent of the 10 indicates how many places from the decimal the number starts. This form is called scientific notation. It is always written with a number between 1 & 10 (called the coefficient) times 10 raised to some integer power (called the order of magnitude). Ex) convert to standard notation: 3.2 x 10 6 Ex) convert to standard notation: 5.24 x 10-3 Ex) convert to scientific notation: Ex) convert to scientific notation: Scientific notation, multiplying and dividing Calculate and write in proper scientific notation Ex) (3 x 10 4 ) x (2 x 10-7 ) Ex) Ex) (4 x 10 3 ) x (6 x 10 5 )

71 Note that after the initial calculation, the coefficient is not between 1 & 10. We need to break out or shift powers of 10 so that it is. One way to do that is to rewrite the coefficient in scientific notation, then combine the two powers of 10. Ex)

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