WEBVTT mathematics/algebra-2/eaton 00:00:00.500 --> 00:00:01.900 Welcome to Educator.com. 00:00:01.900 --> 00:00:08.300 Today is our first lesson for the Algebra II series, and we are going to start out with some review of concepts from Algebra I. 00:00:08.300 --> 00:00:14.600 If you need more detail about any of these concepts, please check out the Algebra I series here at Educator. 00:00:14.600 --> 00:00:20.200 The first session is on expressions and formulas. 00:00:20.200 --> 00:00:25.000 Recall the earlier concepts of variables and algebraic expressions: 00:00:25.000 --> 00:00:33.700 starting out with some definitions, a variable is a letter or symbol that is used to represent an unknown number. 00:00:33.700 --> 00:00:44.800 It could be any letter; frequently, x, y, and z are used, but you could choose n or s or w. 00:00:44.800 --> 00:00:57.500 Algebraic expressions means that terms using both variables and numbers are combined using arithmetic operations. 00:00:57.500 --> 00:01:05.600 Remember that a term is a number, or a variable, or both. 00:01:05.600 --> 00:01:16.200 So, a term could be 4--that is a constant, and it is a term; it could be 2x; it could be y². 00:01:16.200 --> 00:01:22.200 And when these are combined using arithmetic operations, then they are known as expressions. 00:01:22.200 --> 00:01:25.300 And when variables are involved, then they are algebraic expressions. 00:01:25.300 --> 00:01:37.100 For example, 4x³+2xy-1 would be an example of an algebraic expression. 00:01:37.100 --> 00:01:47.000 The rules specifying order of operations are very important; and they are used in order to evaluate algebraic expressions. 00:01:47.000 --> 00:01:56.200 Recall the procedure to evaluate an expression using the order of operations. 00:01:56.200 --> 00:02:30.800 First, evaluate expressions that are inside grouping symbols: examples would be parentheses, braces, and brackets. 00:02:30.800 --> 00:02:40.100 The next thing, when you are evaluating an algebraic expression, is to evaluate powers. 00:02:40.100 --> 00:02:55.500 So, if a term is raised to a power (such as 4² or 3⁴), you need to evaluate that next; that is the second step. 00:02:55.500 --> 00:03:12.700 Next is to multiply and divide, going from left to right. 00:03:12.700 --> 00:03:18.000 You start out at the left side of an expression; and if you hit something that needs to be divided, you do that. 00:03:18.000 --> 00:03:23.200 And you proceed towards the right; if you see something that needs to be multiplied, you do that. 00:03:23.200 --> 00:03:28.400 It is not "multiply all the way, and then go back and divide"; it is "start at the left; any multiplication or division--do it." 00:03:28.400 --> 00:03:35.800 Move to the next step; move towards the right; multiply or divide...and so on, until all of that has been taken care of. 00:03:35.800 --> 00:03:46.600 Finally, you do the same thing with addition and subtraction: you add and subtract from left to right. 00:03:46.600 --> 00:03:51.000 And we will be illustrating these concepts in the examples. 00:03:51.000 --> 00:03:55.800 One thing to recall is that a fraction bar can function as a grouping symbol. 00:03:55.800 --> 00:04:08.000 For example, if I have something like 3x-2x+2, all over 4(x+3)+3, I would treat this as a grouping symbol. 00:04:08.000 --> 00:04:13.400 And I would simplify this as far as I could, going through my four steps; and then I would simplify this; 00:04:13.400 --> 00:04:19.500 and then I would divide this simplified expression on the top by the simplified expression on the bottom. 00:04:19.500 --> 00:04:25.600 And remember, the reason that we use order of operations is that, if we didn't, and everybody was just doing things their own way, 00:04:25.600 --> 00:04:29.500 we couldn't really communicate using math, because people would write something down, 00:04:29.500 --> 00:04:33.000 and somebody might do it in a different order and come up with a different answer. 00:04:33.000 --> 00:04:40.800 So, this way, it is an agreed-upon set of rules that everyone follows. 00:04:40.800 --> 00:04:48.300 Monomials: a monomial is a product of a number and 0 or more variables. 00:04:48.300 --> 00:05:04.300 Again, refreshing your memory from Algebra I: examples of a monomial would be 5y, 6xy², z, 5. 00:05:04.300 --> 00:05:07.200 So, it says it is a product of a number and zero or more variables. 00:05:07.200 --> 00:05:14.200 Here, there aren't any variables, so that actually is simply a constant; but it is still called a monomial, also. 00:05:14.200 --> 00:05:21.100 Here, I have 5 times one variable; here I have multiple variables. 00:05:21.100 --> 00:05:28.000 This is examples of...these are all monomials. 00:05:28.000 --> 00:05:46.000 A constant is simply a number; so, it could be -3 or 6 or 14; those are constants. 00:05:46.000 --> 00:06:03.500 Coefficients: a coefficient is the number in front of the variable. 00:06:03.500 --> 00:06:12.500 Up here, I said I had 5y and 6xy²: this is a coefficient: 5 is a coefficient, and 6 is a coefficient. 00:06:12.500 --> 00:06:18.300 When you see something like z, it does have a coefficient: it actually has a coefficient of 1. 00:06:18.300 --> 00:06:25.700 However, by convention, we usually don't write the 1--we just write it as z, but it actually does have a coefficient of 1. 00:06:25.700 --> 00:06:34.300 Next, degree--the degree of a monomial: the degree of a monomial is the sum of the degrees of all of the variables. 00:06:34.300 --> 00:06:48.100 So, it is the sum of the degree of all of the variables. 00:06:48.100 --> 00:06:57.700 For example, 3xy²z⁴: if I want to find the degree, I am going to add the degree of each variable. 00:06:57.700 --> 00:07:09.300 This is x (but that really means x to the 1--the 1 is unstated) plus y² (the degree is 2), plus z⁴ (the degree is 4). 00:07:09.300 --> 00:07:15.300 Adding these up, the degree for this monomial is 7. 00:07:15.300 --> 00:07:27.600 When we talk about powers: powers refer to a number or variable being multiplied by itself n times, where n is the power. 00:07:27.600 --> 00:07:35.500 For example, if I say that I have 5², what I am really saying is 5 times 5. 00:07:35.500 --> 00:07:40.800 So, 5 is being multiplied by itself twice, where n equals 2. 00:07:40.800 --> 00:07:57.100 I could say I have y⁴: that equals y times y times y times y; and here, the power is 4. 00:07:57.100 --> 00:08:05.600 OK, continuing on with more concepts: a polynomial is a monomial or a sum of monomials. 00:08:05.600 --> 00:08:12.000 Recall the concepts of term, like terms, binomial, and trinomial. 00:08:12.000 --> 00:08:16.600 A polynomial is simply an expression in which the terms are monomials. 00:08:16.600 --> 00:08:24.300 And we say "sum," but this applies to subtraction, as well--a polynomial can certainly involve subtraction. 00:08:24.300 --> 00:08:39.100 For example, 4x²+x or 2y²+3y+4: these are both polynomials. 00:08:39.100 --> 00:08:52.400 We also could say that 5z is a polynomial, but it is also a monomial; there is only one term, so it is a polynomial, but we usually just say it is a monomial. 00:08:52.400 --> 00:09:03.900 OK, so looking at these other words: a binomial is a polynomial that contains two terms. 00:09:03.900 --> 00:09:12.200 So, it is the sum of two monomials, whereas the trinomial is the sum of three monomials. 00:09:12.200 --> 00:09:21.400 A monomial is simply a single monomial. 00:09:21.400 --> 00:09:32.000 Recall that, as discussed, a term is a number or a letter (which is a variable) or both, separated by a sign. 00:09:32.000 --> 00:09:42.600 Terms could be a number; they could be a variable; or they could be both. 00:09:42.600 --> 00:09:54.200 3x-7+z: here, I have a number and a variable; here, I just have a number (I have a constant); here, I just have a variable. 00:09:54.200 --> 00:10:02.400 And they are separated by signs--by a negative sign and a positive sign--so each one of these is a term. 00:10:02.400 --> 00:10:09.600 The concept of like terms is very important, because like terms can be added or subtracted. 00:10:09.600 --> 00:10:25.800 Like terms contain the same variables to the same powers. 00:10:25.800 --> 00:10:35.700 For example, 1 and 6 are like terms; they don't contain any variables, so they are like terms. 00:10:35.700 --> 00:10:45.100 3xy and 4xy are like terms; they both contain an x to the first power and a y to the first power. 00:10:45.100 --> 00:10:53.500 2y² and 8y² are also like terms: they both contain a y raised to the second power. 00:10:53.500 --> 00:11:01.300 And so, these can be combined: they can be added and subtracted. 00:11:01.300 --> 00:11:10.500 A formula is an equation involving several variables (2 or more), and it describes a relationship among the quantities represented by the variables. 00:11:10.500 --> 00:11:17.900 And we have worked with formulas previously: and just to review, one formula that we talked about is the Pythagorean Theorem. 00:11:17.900 --> 00:11:23.600 That is a²+b²=c², where c is the length 00:11:23.600 --> 00:11:29.400 of the hypotenuse of a right triangle, and a and b are the lengths of the two sides. 00:11:29.400 --> 00:11:37.300 And this tells us the relationship among the three sides of the triangle. 00:11:37.300 --> 00:11:46.200 And that is really what formulas are all about, and really what algebra is all about: describing relationships between various things. 00:11:46.200 --> 00:11:50.700 And of course, during this course, we are going to be working with various formulas. 00:11:50.700 --> 00:11:58.000 OK, in this example, we are asked to simplify or evaluate an algebraic expression. 00:11:58.000 --> 00:12:04.200 5x²...and they are telling me x=3, y=-3; so I have some x terms and some y terms. 00:12:04.200 --> 00:12:16.400 My first step is to substitute: so, everywhere I have an x, I am putting in a 3; everywhere I have a y, I am putting in a -3. 00:12:16.400 --> 00:12:20.600 So, here I have 3xy, so here it is going to be 3 times 3 times -3. 00:12:20.600 --> 00:12:26.300 Recall the order of operations: the first thing I am going to do is to get rid of the grouping symbols. 00:12:26.300 --> 00:12:34.900 Take care of the parentheses; and looking, I do have parentheses. 00:12:34.900 --> 00:12:45.900 In here, I have a negative and a negative; so I am simplifying that just to positive 3. 00:12:45.900 --> 00:12:55.100 OK, continuing to simplify inside the parentheses: 3 plus 3 is 6. 00:12:55.100 --> 00:12:57.800 I completed my first step in the order of operations. 00:12:57.800 --> 00:13:04.900 The next thing to do is evaluate powers; and I do have some terms that are raised to various powers. 00:13:04.900 --> 00:13:20.700 3² is 9, minus 2 times 6³; so, 6 times 6 is 36, times 6 is 216. 00:13:20.700 --> 00:13:27.500 That took care of my powers; and the next thing is going to be to multiply and divide. 00:13:27.500 --> 00:13:32.400 And when we do that, we always proceed from left to right. 00:13:32.400 --> 00:13:45.100 2 times 216 is 432; OK, now I have: 3 times 3 is 9; 9 times -3 is -27. 00:13:45.100 --> 00:13:50.200 I am going to rewrite this as 9 minus 432 minus 27. 00:13:50.200 --> 00:13:59.100 Finally, add and subtract; and this is going from left to right. 00:13:59.100 --> 00:14:13.800 9 minus 432 gives me -423, minus 27 (so now I have another bit of subtraction to do--that is -423-27) gives me -450. 00:14:13.800 --> 00:14:19.700 So again, the first step was substituting in 3 and -3 for x and y. 00:14:19.700 --> 00:14:27.200 The next step was to get rid of my grouping symbols; evaluate the powers; 00:14:27.200 --> 00:14:32.700 multiply and divide, going from left to right (and I just had multiplication here); 00:14:32.700 --> 00:14:39.000 and then, add and subtract, going from left to right, to get -450. 00:14:39.000 --> 00:14:46.400 In this second example, again, we are asked to evaluate an algebraic expression; and here, we have three variables: a, b, and c. 00:14:46.400 --> 00:15:07.100 So, carefully substituting in each of these, a=-1...so -1², minus 2, times b (b is 2), times c (c is 3), plus 3³. 00:15:07.100 --> 00:15:18.600 Here, I have c² in the denominator; so that gives me 3², minus 2, times a (which is -1), times b (which is 2). 00:15:18.600 --> 00:15:22.700 Since there was a lot of substituting, it is a good idea to check your work. 00:15:22.700 --> 00:15:33.800 a is -1 (that is -1²), minus 2, times b, times c, plus c³; 00:15:33.800 --> 00:15:40.700 all of that is divided by 3² (so that is c²) minus 2, times a, times b. 00:15:40.700 --> 00:15:45.200 Everything looks good; now, the first thing I want to do is eliminate grouping symbols. 00:15:45.200 --> 00:15:50.700 Recall that, in this type of a case, the fraction bar is functioning as a grouping symbol. 00:15:50.700 --> 00:15:58.200 So, the whole numerator should be simplified, and the denominator should be simplified; and then I should divide one by the other. 00:15:58.200 --> 00:16:04.300 Starting with the numerator: within the numerator, there are not any grouping symbols, 00:16:04.300 --> 00:16:11.000 so I am going to go ahead and go to the next step, which is to take care of powers. 00:16:11.000 --> 00:16:31.500 And -1 times -1 is 1; and then, I have 3³; that is 3 times 3 (is 9), times 3 (is 27). 00:16:31.500 --> 00:16:36.800 OK, I can do the same thing in the denominator; I can just do these both in parallel. 00:16:36.800 --> 00:16:40.400 And so, I am going to evaluate the powers in the denominator. 00:16:40.400 --> 00:16:45.600 3 times 3 is 9; and then, I don't have any more powers--OK. 00:16:45.600 --> 00:16:56.300 So, I took care of that; my next step is going to be to multiply and divide. 00:16:56.300 --> 00:17:14.900 OK, so I have, in the numerator, 1 minus 2 times 2 (is 4), and then 4 times 3 (is 12), plus 27. 00:17:14.900 --> 00:17:26.300 So, that took care of that step; now, in the denominator, I have 9, and then I have minus 2, times -1. 00:17:26.300 --> 00:17:42.800 So, 2 times -1 is going to give me -2; -2 times 2 is going to give me -4. 00:17:42.800 --> 00:17:47.700 OK, so now, I have taken care of all of the multiplication and the division. 00:17:47.700 --> 00:17:53.100 The next step is to add and subtract--once again, going from left to right. 00:17:53.100 --> 00:18:05.700 So, starting up here, the next step is going to be 1 minus 12; 1 minus 12 is going to give me -11. 00:18:05.700 --> 00:18:12.000 So, it is -11 plus 27; that is going to leave me with 16 in the numerator. 00:18:12.000 --> 00:18:19.000 In the denominator, I have 9, minus -4; well, a negative and a negative gives me a positive, 00:18:19.000 --> 00:18:29.500 so in the denominator, I actually have 9 plus 4, which gives me 13. 00:18:29.500 --> 00:18:31.800 The result is 16 over 13. 00:18:31.800 --> 00:18:40.300 Again, starting out by substituting values for a, b, and c...I have done that in this first step. 00:18:40.300 --> 00:18:47.800 And then, I treat this fraction bar as a big grouping symbol, and then I take care of the numerator and the denominator separately. 00:18:47.800 --> 00:18:51.900 You could have done them one at a time, or you can do steps at the same time. 00:18:51.900 --> 00:18:58.600 So, first, evaluate the powers; I did that in the numerator; I did that in the denominator (I am treating them separately). 00:18:58.600 --> 00:19:05.700 Multiplying and dividing: I did my multiplication here and in the denominator. 00:19:05.700 --> 00:19:12.100 And finally, adding and subtracting to get 16/13. 00:19:12.100 --> 00:19:18.600 Example 3: The formula for the area of a triangle is Area equals 1/2 bh. 00:19:18.600 --> 00:19:24.400 So, this is actually that the area equals one-half the base times the height of the triangle. 00:19:24.400 --> 00:19:32.900 Find the height if a is 32, and the base (b) is 8. 00:19:32.900 --> 00:19:39.100 OK, here we are being asked to find the height, and we are given the other two variables. 00:19:39.100 --> 00:19:43.400 So, let me rewrite the formula: area equals 1/2 base times height. 00:19:43.400 --> 00:19:45.400 Now, I am going to substitute in what I was given. 00:19:45.400 --> 00:19:54.900 I am given the area; I am given the base; and I need to find the height. 00:19:54.900 --> 00:20:06.300 What I need to do is isolate h; so, first simplifying this: 32=...well, 1/2 of 8 is 4, so that gives me...4h. 00:20:06.300 --> 00:20:16.200 Next, divide both sides of the equation by 4 (32/4 and 4h/4) in order to isolate that. 00:20:16.200 --> 00:20:28.200 Well, 32 divided by 4 is 8; the 4's cancel out on the right; and then just rewriting this in a more standard form, with the variable on left, the height is 8. 00:20:28.200 --> 00:20:36.700 So again, first just write out the formula; substitute in a and b (which I was given). 00:20:36.700 --> 00:20:42.400 And then, solve for the height. 00:20:42.400 --> 00:20:50.000 The temperature in Fahrenheit is F=9/5C+32, where C is the temperature in Celsius. 00:20:50.000 --> 00:20:55.500 If the temperature is 78 degrees Fahrenheit, what is it in Celsius? 00:20:55.500 --> 00:21:02.800 Rewrite the formula and substitute in what is given. 00:21:02.800 --> 00:21:07.000 What is given is that the temperature in Fahrenheit is 78. 00:21:07.000 --> 00:21:12.000 And I am looking for Celsius (I always keep in mind what I am looking for--what is my goal?). 00:21:12.000 --> 00:21:21.400 And that is +32; my goal here is going to be to solve for C--to isolate that. 00:21:21.400 --> 00:21:30.100 Subtracting 32 from both sides gives me 46=9/5C. 00:21:30.100 --> 00:21:42.600 Now, in order to isolate the Celsius, I am going to multiply both sides by 5/9. 00:21:42.600 --> 00:21:50.400 When I do that, I am going to get 5 times 46 (is 230), and that is divided by 9. 00:21:50.400 --> 00:21:57.200 Here, that all cancels out; so, rewriting this, Celsius equals 230/9. 00:21:57.200 --> 00:22:07.600 That is not usually how we talk about temperature; so simplifying that, if I took 230 and divided it by 9, I would get approximately 25.5 degrees Celsius. 00:22:07.600 --> 00:22:15.300 So again, the formula for converting Fahrenheit into Celsius (or vice versa) is given. 00:22:15.300 --> 00:22:26.000 I substituted in 78 degrees and figured this out: so, 78 degrees would be equal to approximately 25.5 degrees Celsius. 00:22:26.000 --> 00:22:23.000 That concludes today's lesson on Educator.com; and I will see you again for the next lesson.