The student solutions shown below highlight the most commonly used approaches and also some that feature nice use of algebraic polynomial formulas.
|
|
- Randolf Cameron
- 5 years ago
- Views:
Transcription
1 Print Assign Submit Solution and Commentary Online Resources Scoring Rubric [pdf] Teacher Packet [pdf] Strategy 11: Get Unstuck Strategy Examples [pdf] Polynomial Power [Problem #5272] Comments and Sample Solutions The key to solving this problem was taking the x + 1/x and treating it as a binomial. It could then be raised to a power, or multiplied by itself, to increase the powers of x. But whether raising to powers or multiplying, don't forget that there are middle terms to be considered! The student solutions shown below highlight the most commonly used approaches and also some that feature nice use of algebraic polynomial formulas. For the first question, almost everyone either squared the (x + 1/x) or multiplied it by itself. Some students, including Hari Vasireddy and Melodie Yu, used the formula that (a + b) 2 = a 2 + 2ab + b 2 to help with the squaring process. For the second question, the two most common methods were to either multiply three (x + 1/x) binomials together or to take the result of the first question and multiply that by another (x + 1/x) since it already represents two of them multiplied together. Melodie again used a neat formula for the sum of cubes, which says that a 3 + b 3 = (a + b)(a 2 - ab + b 2. By using x for a and 1/x for b she was able to quickly find x 3 + 1/x 3. In both problems, while the terms on the ends gave you the pieces you wanted, the terms in the middle cleaned up nicely and were able to be replaced with numeric values. It was pretty neat the way that worked! Another handy formula for the second problem shows the result of cubing a binomial: (a + b) 3 = a 3 + 3a 2 b + 3ab 2 + b 3. The coefficients of this expression, , are linked to Pascal's Triangle, and that's something that a number of people picked up on for the Extra question. Edel Auh and Rebecca Dun both talk about using Pascal's Triangle in their work, and Rebecca does a great job of explaining how it works in this context. Edel makes the excellent comment that in order to use it to find something like x 9 + 1/x 9 you need to know the values of at least some of the earlier expressions in the progression. Other people calculated several steps of the progression and then noticed a pattern. They saw that to find the value of a given step, it turned out to be five times the value from two steps before minus the value from one step before. That would certainly let you calculate the value of higher stages quickly as you could skip all the algebra and just use the results. But again, it requires knowing earlier values to reach the one you are trying to calculate. Can anyone find a way to go directly to the answer you want without having to know earlier values? Congratulations to Edel, Fatma, Liu, Haripriya, Rebecca, Melodie, and everyone else who solved this challenging problem. Nice work! - Riz From: Xingcheng L, age 13, Rancho San Joaquin Middle School x 2 + 1/x 2 would be equal to 23. x 3 + 1/x 3 is equal to 110. To find out what x 2 + 1/x 2 is, I multiplied the first equation, which was x + 1/x = 5, by itself. (x + 1/x)(x + 1/x) = 5*5 Page 1 of 10
2 Using the foil method, I quickly simplified and solved the equation to find what x 2 + 1/x 2 is. (x + 1/x)(x + 1/x) = 5*5 x 2 + x/x + x/x+ 1/x 2 = 25 x /x 2 = 25 x 2 + 1/x 2 = 23 To find out what x 3 + 1/x 3 is, I multiplied the first equation and the second one together. (x 2 + 1/x 2 )(x+1/x) = 23*5 x 3 + x + 1/x + 1/x 3 = 115 Then, I saw that there was an x + 1/x in the middle of the equation, and this could be replaced with 5. Then, you solve the equation. x /x 3 = 115 x 3 + 1/x 3 = 110 Reflection This POW was pretty easy, because all I had to do was multiply some equations together. At first, I thought that you were supposed to solve the problem by finding what x is, but after a while, I got the feeling that that would be hard, so I decided to just multiply the equations together. However, the extra was a little challenging because there was no real answer. Also, I only came to my pattern by Page 2 of 10
3 chance. Extra The pattern that I found was that x n + 1/x n = 5(x n-1 + 1/x n-1 ) - (x n-2 + 1/x n-2. However, this does not apply to the first and second equations. To find the pattern, I first found out what all the expressions up to x 7 + 1/x 7. x+1/x = 5 x 2 + 1/x 2 = 23 x 3 + 1/x 3 = 110 x 4 + 1/x 4 = 527 x 5 + 1/x 5 = 2525 x 6 + 1/x 6 = x 7 + 1/x 7 = Then, I got frustrated because I did not find a pattern, so I just did something random. I wanted to find how I could change into I just picked a random number, which turned out to be 5, and multiplied by 5. I got as the product. I then subtracted 2525 from 60490, and surprisingly, I got I tried this pattern with all of the other equations, and they all worked except for the first two. If you want to find the nth number in the sequence, then you multiply the previous number by 5 and then subtract the next previous one. This can also be said as x n + 1/x n = 5(x n-1 + 1/x n- 1 ) (x n-2 + 1/x n-2 ) Page 3 of 10
4 From: Haripriya V, age 13, Rancho San Joaquin Middle School x squared + 1/x squared is 23. x cubed + 1/x cubed is 110. EXTRA: (see explanation) In order to find the answer to x 2 + 1/x 2 I used the formula (a + b) 2 = a 2 + b 2 + 2ab. a = x b = 1/x (a + b) 2 = a 2 + b 2 + 2ab (x + 1/x) 2 = x 2 + 1/x (x + 1/x) = 5, so I replace that part of the equation with 5. (5) 2 = x 2 + 1/x = x 2 + 1/x 2 x squared + 1/x squared is 23. I calculated (x 2 + 1/x 2 )(x + 1/x). x 2 + 1/x 2 = 23 x + 1/x = 5 I multiply both values using the Distributive Property. (x 2 + 1/x 2 )(x + 1/x) x 3 + x + 1/x + 1/x 3 x 3 + 1/x 3 + (x + 1/x) = 23 times 5 x 3 + 1/x = 23 times 5 x 3 + 1/x 3 = 23 times 5-5 = EXTRA: After performing trial and error with exponents following the given pattern, I (with the help of my mother) was able to come up with a common equation that would work for the problems, excluding x + 1/x and x 2 + 1/x 2. x n + 1/x n = (x n-1 + 1/x n-1 )(x + 1/x) - (x n-2 + 1/x n-2 ) nth Term = (n-1)th Term x 5 - (n-2)th Term. Page 4 of 10
5 n=1 x + 1/x = 5 = 5 n=2 x 2 + 1/x 2 = 23 = 23 n=3 x 3 + 1/x 3 = 23x5-5 = 110 n=4 x 4 + 1/x 4 = 110x5-23 = 527 n=5 x 5 + 1/x 5 = 527x5-110 = 2,525 n=6 x 6 + 1/x 6 = 2525x5-527 = 12,098 n=7 x 7 + 1/x 7 = 12098x = 57,965 n=8 x 8 + 1/x 8 = 57965x = 277,727 n=9 x 9 + 1/x 9 = x =1,330,670 REFLECTION: It was challenging to figure out the future patterns, and required persistence. I enjoyed it, however, when I found out the method to finding the answer of any of that kind of problem. My mother helped me figure out the equation that would work for the extra. It was helpful when found out I could use the values from previous equations to figure out the answers for new ones. Being able to "split" large equations and group them using a parenthesis was also a plus. From: Rebecca D, age 13, Rancho San Joaquin Middle School The values for x^2 + 1/x^2 is 23 and x^3 + 1/x^3 is 110. Extra Credit: I saw that Pascal's Triangle showed the coffecients of the equations and that to find x^9 + 1/x^9, you only needed to find what the odd power equations were equal to which he First I squared x + 1/x = 5 to see if it equals x^2 + 1/x^2. (x + 1/x)^2 = (5)^2 x^ /x^2 = 25 Once I saw this, I subracted 2 from 25 in order to find what x^2 + 1/x^2 is equal to. x^2 + 1/x^2 = 23 Next, I cubed x + 1/x = 5 to see if it is close to x^3 + 1/x^3. (x + 1/x)^3 = (5)^3 x^3 + 3x + 3/x + 1/x^3 = 125 At first, this looked confusinng, but I saw this and substituted to find what x^3 + 1/x^3 is equal to. x^3 + 3(x + 1/x) + 1/x^3 = 125 x^3 + 3(5) + 1/x^3 = 125 x^ /x^3 = 125 x^3 + 1/x^3 = 110 So that's how I found x^2 + 1/x^2 = 23 and x^3 + 1/x^3 = 110. Page 5 of 10
6 Extra Credit: The first pattern I noticed was that this problem seemed to connect to Pascal's Triangle because it felt awfully familar to the POW, Sierpinski Stages, and what my teacher, Mr. Compton, was saying about this triangle pattern connecting to the coefficients of some equations. So then I looked up a picture of Pascal's Triangle and saw, just like Mr. Compton explained, that the triangle showed the cofficients of the equations. So then I tried x^5 + 1/x^5 and (x + 1/x)^5 to see if really does work. (x + 1/x)^5 = (5)^5 x^5 + 5x^3 + 10x + 10(1/x) + 5(1/x^3) + 1/x^5 = 3,125 x^5 + 5(x^3 + 1/x^3) + 10(x + 1/x) + 1/x^5 = 3,125 x^5 + 5(110) + 10(5) + 1/x^5 = 3,125 x^ /x^5 = 3,125 x^5 + 1/x^5 = 2,525 When I did this, I realized I could just find the values of the odd power equations to find x^9 + 1/x^9, so then I found the value of x^7 + 1/x^7 with (x + 1/x)^7. (x + 1/x)^7 = (5)^7 x^7 + 7x^5 + 21x^3 + 35x + 35(1/x) + 21(1/x^3) + 7(1/x^5) + 1/x^7 = 78,125 x^7 + 7(x^5 + 1/x^5) + 21(x^3 + 1/x^3) + 35(x + 1/x) + 1/x^7 = 78,125 x^7 + 7(2,525) + 21(110) + 35(5) + 1/x^7 = 78,125 x^7 + 17, , /x^7 = 78,125 x^7 + 1/x^7 = 57,965 Last but not least, I found the value of x^9 + 1/x^9 with (x + 1/x) ^9. (x + 1/x)^9 = (5)^9 x^9 + 9x^7 + 36x^5 + 84x^ x + 126(1/x) + 84(1/x^3) + 36(1/x^5) + 9(1/x^7) + 1/x^9 = 1,953,125 x^9 + 9(x^7 + 1/x^7) + 36(x^5 + 1/x^5) + 84(x^3 + 1/x^3) + 126(x + 1/x) + 1/x^9 = 1,953,125 x^9 + 9(57,965) + 36(2,525) + 84(110) + 126(5) + 1/x^9 = 1,953,125 x^ , , , /x^9 = 1,953,125 x^9 + 1/x^9 = 1,330,670 So that's the patterns and methods I used to make it easier to find the value of x^9 + 1/x^9, which is 1,330,670. Reflection: I would like to thank a lot of people for helping me on POW. First, I would like to thank Khue for showing me that finding x isn't necessary. Also, I would like to thank Jim for telling me that he used a equation (he didn't tell me what it was until after i solved it) because it was what lead me to the idea of Pascal's Triangle. Next, I would like to thank Mr. Compton for giving away all those hints about the POW, like how to solve x^2 + 1/x^2. Last, I would like to thank my mom for showing me an example of the process of how the triangle and these equations are linked (x + 1/x are the same coefficients as row 2 in the triangle). I found these answers reasonable because for x^2 + 1/x^2 and x^3 + 1/x^3, I knew they had to be close to 25 and 125. Also, for the Page 6 of 10
7 extra, I knew the value of x^7 + 1/x^7 was going to be a big number, x^9 + 1/x^9 had to be bigger which it is. The basic process of this POW is raise x + 1/x = 5 to whatever the power they ask and then find all the values inbetween and then subtract all those values from 5 to the something power and that will be the value. To check my answer for values, I looked over all my work and doublechecked to make sure I got the right answer. I also asked other kids at school if they got the same answer and they did. I found this POW very related to another POW, Sierpinski Stages, becuase they found have to do with Pascal's Triangle and some mysterious equation that really hard to find. From: Edel A, age 12, Rancho San Joaquin Middle School x^2 + 1/x^2 is equal to 23 and x^3 + 1/x^3 is equal to 110. Extra: One way is by using the formula x^n + 1/x^n = 5(x^(n-1) + 1/x^ (n-1)) - (x^(n-2) + 1/x^(n-2)). Using Pascal's Triangle is another way to solve higher value expressions. I decided to solve x^2 + 1/x^2 first. I realized that I could find (x^2 + 1/x^2) by squaring (x + 1/x) and then simplifying. x + 1/x = 5 (x + 1/x)^2 = 5^2 (x + 1/x)(x + 1/x) = 25 x^ /x^2 = 25 x^2 + 1/x^2 = 23 Therefore, I reached the conclusion that (x^2 + 1/x^2) is equal to 23. Next, I solved for (x^3 + 1/x^3). I realized that I could cube (x + 1/x) find out what (x^3 + 1/x^3) is equal to. (x + 1/x)(x + 1/x)(x + 1/x) = 5^3 (x^ /x^2)(x + 1/x) = 125 (x^3 + 2x + 1/x) + (x + 2/x + 1/x^3) = 125 (x^3 + 3x + 3/x + 1/x^3 = 125 (x^3 + 1/x^3) + (3x + 3/x) = 125 (x^3 + 1/x^3) + 3(x + 1/x) = 125 (x^3 + 1/x^3) + 3(5) = 125 (x^3 + 1/x^3) + 15 = 125 (x^3 + 1/x^3) = 110 Therefore, (x^3 + 1/x^3) is equal to 110. Extra: A method to finding out a higher value expression is by using the formula: x^n + 1/x^n = 5(x^n-1 + 1/x^n-1) - (x^n-2 + 1/x^n-2) For example: Solve: x^9 + 1/x^9 Need to know (x^(n-1) + 1/x^(n-1)) and (x^(n-2) + 1/x^(n-2)) which Page 7 of 10
8 is (x^8 + 1/x^8) and (x^7 + 1/x^7). 5(x^8 + 1/x^8) - (x^7 + 1/7) = (x^9 + 1/x^9) 5(277727) - (57965) = (x^9 + 1/x^9) = x^9 + 1/x^9 The other method that I found is related to Pascal's Triangle. I noticed that when I expand (x + 1/x)^n, the coefficients of the expanded equation were equal to the first half of the numbers in the nth row of Pascal's Triangle. They are coefficients to expressions in parantheses. The first expression is (x^n + 1/x^n). For each consecutive expression, the n decreases by two. This complete expression would be equal to 5^n since (x + 1/x) is equal to 5. After writing this new equation, I just needed to fill in the values and multiply to get my answer. Example: Solving for (x^9 + 1/x^9) 9th row of Pascal's Triangle (from left to middle): (x^9 + 1/x^9) + 9(x^7 + 1/x^7) + 36(x^5 + 1/x^5) + 84(x^3 + 1/x^3) + 126(x + 1/x) = 5^9 (x^9 + 1/x^9) + 9(57965) + 36(2525) + 84(110) + 126(5) = (x^9 + 1/x^9) + (521685) + (90900) + (9240) + (630) = (x^9 + 1/x^9) = (x^9 + 1/x^9) = The downside for both of these methods is that you need to know what other values of n in (x^n + 1/x^n) are equal to in order to use these methods. In reality, there is a lot more work, such as finding out (x^7 + 1/x^7), involved Reflection: This Problem of the Week was a bit difficult. I had help from adults on solving for (x^2 + 1/x^2), but I had to solve (x^3 + 1/x^3) on my own. At first, I thought that the equation I used above for solving (x^3 + 1/x^3) didn't work, but with the help of my tutor, I was able to find out that not only did it work, it gave me the correct answer. Then, for the Extra, I had help finding out the formula from my tutor and the method using Pascal's triangle from Mr.Compton. I didn't even think about looking at Pascal's Triangle, but once he explained it to me, I was able to understand how Pascal's Triangle was related to the problem. I was also able to simplify the method even more to discover an easier and quicker method. However when solving for (x^9 + 1/x^9), I got confused on what the number on the right side of the equation had to be, but my tutor helped me discover that it equaled to 5^9 and explain it. I would like to thank my tutor and my dad for helping me out and proofreading my explanation. I would also like to thank Mr. Compton and Brian S. for helping me out on the Extra. All in all, this Problem of the Week was a bit challenging, but I enjoyed it because it fits in with what we're learning in class right now. I think this Problem of the Week helped me learn more about polynomials and the way they work. I got to use FOIL, something I enjoy, substitution, and set up equations. Solving this was a lot of fun and I learned a lot. Page 8 of 10
9 From: Fatma., age 17, Gesamtschule Hattingen The value of the term x²+ 1/x² is 23. The value of the term x³+ 1/x³ is x²+1/x²=? I know that x+1/x=5 (x+1/x)²=5² x²+2*x*1/x+1/x²= 25 x²+2+1/x²= 25 x²+1/x²= 23 I square both sides I use the binomial formular I multiply the inner term I substact 2 on both sides 2. x³+1/x³=? (x+1/x)³=5³ I cube both sides (x+1/x)(x+1/x)(x+1/x)=125 I multiply the the fist two brackets (x²+2+1/x²)(x+1/x)=125 then i multiply it with the third x³+1x²/x+2x+2/x+1x/x²+1/x³=125 bracked x³+3x+3/x+1/x³=125 x³+3(x+1/x)+1/x³=125 I substitute the bracket for five x³+15+1/x³= I substract 15 on both sides x³+1/x³=110 From: Melodie Y, age 13, Rancho San Joaquin Middle School Answer: The value of x^2 + 1/x^2 is equal to 23 and I got the value of 110 for x^3 + 1/x^3. Explanation: First I noticed that I could use the equation a^2 + 2ab + b^2 to find out the value of x^2 + 1/x^2. I used x for a and 1/x for b. I would take x + 1/x = 5 and square it so I would get x^ /x^2 = 25. Then I would simplify it down to x^2 + 1/x^2 = 23 and that's how I got my first answer. a^2 + 2ab + b^2 (x + 1/x = 5)^2 x^2 + 2(x)(1/x) + (1/x)^2 = 25 x^ /x^2 = 25 x^2 + 1/x^2 = 23 Next to find the value of x^3 + 1/x^3, I used the equation a^3 + b^3 which is equal to (a + b)(a^2 - ab + b^2). I used that equation and used x + 1/x to substitute in. Like the first equation, a = x and b = 1/x. a^3 + b^3 = (a + b)(a^2 - ab + b^2) x + 1/x (x + 1/x)(x^2 - (x)(1/x) + (1/x)^2) Next I simplified it but I noticed that I already knew what x + 1/x was equal to, so I substituted that in for 5. 5(x^ /x^2) 5(x^2 + 1/x^2-1) After I simplified it, I noticed another part of the equation that we had the answer to, x^2 + 1/x^2 which was 23 so I substituted that for 23. Then I simplified and got my answer. 5(23-1) 5(22) 110 Page 9 of 10
10 Extra: I found that I could use the equation a^3 + b^3 = (a + b)(a^2 - ab + b^2) to find the value of the expression x^9 + 1/x^9. You replace a and b with a number that can be simplified in a^9 + 1/b^9. In this case it would be x^3 + 1/x^3. Reflection: I did not get this problem at all. I knew I wasn't supposed to try to solve it because that was the info Mr. Compton gave me but my head still kept trying to solve it. Melody was patient with me and explained it. I was still confused when she explained it but she explained it a little more and I eventually got it. Sidra tried to help me with x^3 + 1/x^3 but it was the wrong answer Drexel University. All Rights Reserved. Page 10 of 10
Solving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationSolving Quadratic & Higher Degree Equations
Chapter 9 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationTo: Amanda From: Daddy Date: 2004 February 19 About: How to solve math problems
to Amanda p.1 To: Amanda From: Daddy Date: 2004 February 19 About: How to solve math problems There are 4 steps in solving the kind of math problem you showed me. I'll list the steps first, then explain
More informationAlgebra. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationMITOCW ocw f99-lec17_300k
MITOCW ocw-18.06-f99-lec17_300k OK, here's the last lecture in the chapter on orthogonality. So we met orthogonal vectors, two vectors, we met orthogonal subspaces, like the row space and null space. Now
More informationPolynomials; Add/Subtract
Chapter 7 Polynomials Polynomials; Add/Subtract Polynomials sounds tough enough. But, if you look at it close enough you ll notice that students have worked with polynomial expressions such as 6x 2 + 5x
More informationMITOCW watch?v=pqkyqu11eta
MITOCW watch?v=pqkyqu11eta The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To
More informationMITOCW MITRES18_005S10_DerivOfSinXCosX_300k_512kb-mp4
MITOCW MITRES18_005S10_DerivOfSinXCosX_300k_512kb-mp4 PROFESSOR: OK, this lecture is about the slopes, the derivatives, of two of the great functions of mathematics: sine x and cosine x. Why do I say great
More information3.4 Pascal s Pride. A Solidify Understanding Task
3.4 Pascal s Pride A Solidify Understanding Task Multiplying polynomials can require a bit of skill in the algebra department, but since polynomials are structured like numbers, multiplication works very
More informationAlgebra & Trig Review
Algebra & Trig Review 1 Algebra & Trig Review This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The
More informationMITOCW watch?v=ko0vmalkgj8
MITOCW watch?v=ko0vmalkgj8 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To
More informationTo factor an expression means to write it as a product of factors instead of a sum of terms. The expression 3x
Factoring trinomials In general, we are factoring ax + bx + c where a, b, and c are real numbers. To factor an expression means to write it as a product of factors instead of a sum of terms. The expression
More informationEQ: How do I identify exponential growth? Bellwork:
EQ: How do I identify exponential growth? Bellwork: 1. Bethany's grandmother has been sending her money for her birthday every year since she turned 1. When she was one, her grandmother sent her $5. Every
More information6: Polynomials and Polynomial Functions
6: Polynomials and Polynomial Functions 6-1: Polynomial Functions Okay you know what a variable is A term is a product of constants and powers of variables (for example: x ; 5xy ) For now, let's restrict
More informationMITOCW ocw f99-lec30_300k
MITOCW ocw-18.06-f99-lec30_300k OK, this is the lecture on linear transformations. Actually, linear algebra courses used to begin with this lecture, so you could say I'm beginning this course again by
More informationAlgebra II Chapter 5: Polynomials and Polynomial Functions Part 1
Algebra II Chapter 5: Polynomials and Polynomial Functions Part 1 Chapter 5 Lesson 1 Use Properties of Exponents Vocabulary Learn these! Love these! Know these! 1 Example 1: Evaluate Numerical Expressions
More informationLesson Plan by: Stephanie Miller
Lesson: Pythagorean Theorem and Distance Formula Length: 45 minutes Grade: Geometry Academic Standards: MA.G.1.1 2000 Find the lengths and midpoints of line segments in one- or two-dimensional coordinate
More informationPre-Algebra PoW Packet Anh s Code September 13, 2010
Pre-Algebra PoW Packet Anh s Code September 13, 2010 http://mathforum.org/pows/ Welcome! Standards The Problem This packet contains a copy of the problem, the answer check, our solutions, teaching suggestions
More informationSolving Quadratic & Higher Degree Equations
Chapter 7 Solving Quadratic & Higher Degree Equations Sec 1. Zero Product Property Back in the third grade students were taught when they multiplied a number by zero, the product would be zero. In algebra,
More informationPolynomials: Add and Subtract
GSE Advanced Algebra Operations with Polynomials Polynomials: Add and Subtract Let's do a quick review on what polynomials are and the types of polynomials. A monomial is an algebraic expression that is
More informationMITOCW ocw f99-lec09_300k
MITOCW ocw-18.06-f99-lec09_300k OK, this is linear algebra lecture nine. And this is a key lecture, this is where we get these ideas of linear independence, when a bunch of vectors are independent -- or
More informationNote: Please use the actual date you accessed this material in your citation.
MIT OpenCourseWare http://ocw.mit.edu 18.06 Linear Algebra, Spring 2005 Please use the following citation format: Gilbert Strang, 18.06 Linear Algebra, Spring 2005. (Massachusetts Institute of Technology:
More informationExtending the Number System
Analytical Geometry Extending the Number System Extending the Number System Remember how you learned numbers? You probably started counting objects in your house as a toddler. You learned to count to ten
More informationCalculus II. Calculus II tends to be a very difficult course for many students. There are many reasons for this.
Preface Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my class notes they should be accessible to anyone wanting to learn Calculus
More informationAlgebra II Polynomials: Operations and Functions
Slide 1 / 276 Slide 2 / 276 Algebra II Polynomials: Operations and Functions 2014-10-22 www.njctl.org Slide 3 / 276 Table of Contents click on the topic to go to that section Properties of Exponents Review
More information3.4 Pascal s Pride. A Solidify Understanding Task
3.4 Pascal s Pride A Solidify Understanding Task Multiplying polynomials can require a bit of skill in the algebra department, but since polynomials are structured like numbers, multiplication works very
More informationImplicit Differentiation Applying Implicit Differentiation Applying Implicit Differentiation Page [1 of 5]
Page [1 of 5] The final frontier. This is it. This is our last chance to work together on doing some of these implicit differentiation questions. So, really this is the opportunity to really try these
More informationMITOCW ocw-18_02-f07-lec17_220k
MITOCW ocw-18_02-f07-lec17_220k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.
More informationModule 3 Study Guide. GCF Method: Notice that a polynomial like 2x 2 8 xy+9 y 2 can't be factored by this method.
Module 3 Study Guide The second module covers the following sections of the textbook: 5.4-5.8 and 6.1-6.5. Most people would consider this the hardest module of the semester. Really, it boils down to your
More informationMath101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2:
Math101, Sections 2 and 3, Spring 2008 Review Sheet for Exam #2: 03 17 08 3 All about lines 3.1 The Rectangular Coordinate System Know how to plot points in the rectangular coordinate system. Know the
More information( )( b + c) = ab + ac, but it can also be ( )( a) = ba + ca. Let s use the distributive property on a couple of
Factoring Review for Algebra II The saddest thing about not doing well in Algebra II is that almost any math teacher can tell you going into it what s going to trip you up. One of the first things they
More informationQuadratic Equations Part I
Quadratic Equations Part I Before proceeding with this section we should note that the topic of solving quadratic equations will be covered in two sections. This is done for the benefit of those viewing
More informationExploring Graphs of Polynomial Functions
Name Period Exploring Graphs of Polynomial Functions Instructions: You will be responsible for completing this packet by the end of the period. You will have to read instructions for this activity. Please
More informationChapter 1 Review of Equations and Inequalities
Chapter 1 Review of Equations and Inequalities Part I Review of Basic Equations Recall that an equation is an expression with an equal sign in the middle. Also recall that, if a question asks you to solve
More information#29: Logarithm review May 16, 2009
#29: Logarithm review May 16, 2009 This week we re going to spend some time reviewing. I say re- view since you ve probably seen them before in theory, but if my experience is any guide, it s quite likely
More informationChapter 1. Making algebra orderly with the order of operations and other properties Enlisting rules of exponents Focusing on factoring
In This Chapter Chapter 1 Making Advances in Algebra Making algebra orderly with the order of operations and other properties Enlisting rules of exponents Focusing on factoring Algebra is a branch of mathematics
More informationSUMMER MATH PACKET. Geometry A COURSE 227
SUMMER MATH PACKET Geometry A COURSE 7 MATH SUMMER PACKET INSTRUCTIONS Attached you will find a packet of exciting math problems for your enjoyment over the summer. The purpose of the summer packet is
More informationGeometry 21 Summer Work Packet Review and Study Guide
Geometry Summer Work Packet Review and Study Guide This study guide is designed to accompany the Geometry Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
More informationChapter 5 Simplifying Formulas and Solving Equations
Chapter 5 Simplifying Formulas and Solving Equations Look at the geometry formula for Perimeter of a rectangle P = L W L W. Can this formula be written in a simpler way? If it is true, that we can simplify
More information6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Tutorial:A Random Number of Coin Flips
6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013 Transcript Tutorial:A Random Number of Coin Flips Hey, everyone. Welcome back. Today, we're going to do another fun problem that
More informationMITOCW watch?v=ed_xr1bzuqs
MITOCW watch?v=ed_xr1bzuqs The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free. To
More informationAlgebra Exam. Solutions and Grading Guide
Algebra Exam Solutions and Grading Guide You should use this grading guide to carefully grade your own exam, trying to be as objective as possible about what score the TAs would give your responses. Full
More informationPOLYNOMIAL EXPRESSIONS PART 1
POLYNOMIAL EXPRESSIONS PART 1 A polynomial is an expression that is a sum of one or more terms. Each term consists of one or more variables multiplied by a coefficient. Coefficients can be negative, so
More informationMITOCW MITRES18_006F10_26_0602_300k-mp4
MITOCW MITRES18_006F10_26_0602_300k-mp4 FEMALE VOICE: The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational
More informationSquaring and Unsquaring
PROBLEM STRINGS LESSON 8.1 Squaring and Unsquaring At a Glance (6 6)* ( 6 6)* (1 1)* ( 1 1)* = 64 17 = 64 + 15 = 64 ( + 3) = 49 ( 7) = 5 ( + ) + 1= 8 *optional problems Objectives The goal of this string
More informationMITOCW ocw f99-lec05_300k
MITOCW ocw-18.06-f99-lec05_300k This is lecture five in linear algebra. And, it will complete this chapter of the book. So the last section of this chapter is two point seven that talks about permutations,
More informationGAP CLOSING. Algebraic Expressions. Intermediate / Senior Facilitator s Guide
GAP CLOSING Algebraic Expressions Intermediate / Senior Facilitator s Guide Topic 6 Algebraic Expressions Diagnostic...5 Administer the diagnostic...5 Using diagnostic results to personalize interventions...5
More informationWe will work with two important rules for radicals. We will write them for square roots but they work for any root (cube root, fourth root, etc.).
College algebra We will review simplifying radicals, exponents and their rules, multiplying polynomials, factoring polynomials, greatest common denominators, and solving rational equations. Pre-requisite
More informationMATHEMATICS TEACHING-RESEARCH JOURNAL ONLINE VOL 9, N 1-2 Summer 2017
Appendix Collection of Aha! Moments collected during the teaching-experiment. FIR TREE Aha! Moment 1. Stage 5 s shape is bigger than the previous one; its shape grows horizontally and follows the pattern
More informationMITOCW R11. Double Pendulum System
MITOCW R11. Double Pendulum System The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for
More informationAlgebra 31 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the ten specific concepts covered in the
More informationbase 2 4 The EXPONENT tells you how many times to write the base as a factor. Evaluate the following expressions in standard notation.
EXPONENTIALS Exponential is a number written with an exponent. The rules for exponents make computing with very large or very small numbers easier. Students will come across exponentials in geometric sequences
More informationBut, there is always a certain amount of mystery that hangs around it. People scratch their heads and can't figure
MITOCW 18-03_L19 Today, and for the next two weeks, we are going to be studying what, for many engineers and a few scientists is the most popular method of solving any differential equation of the kind
More informationRead the text and then answer the questions.
1 Read the text and then answer The young girl walked on the beach. What did she see in the water? Was it a dolphin, a shark, or a whale? She knew something was out there. It had an interesting fin. She
More information25. REVISITING EXPONENTS
25. REVISITING EXPONENTS exploring expressions like ( x) 2, ( x) 3, x 2, and x 3 rewriting ( x) n for even powers n This section explores expressions like ( x) 2, ( x) 3, x 2, and x 3. The ideas have been
More informationMITOCW ocw f99-lec01_300k
MITOCW ocw-18.06-f99-lec01_300k Hi. This is the first lecture in MIT's course 18.06, linear algebra, and I'm Gilbert Strang. The text for the course is this book, Introduction to Linear Algebra. And the
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Concepts Paul Dawkins Table of Contents Preface... Basic Concepts... 1 Introduction... 1 Definitions... Direction Fields... 8 Final Thoughts...19 007 Paul Dawkins i http://tutorial.math.lamar.edu/terms.aspx
More informationIntroduction. So, why did I even bother to write this?
Introduction This review was originally written for my Calculus I class, but it should be accessible to anyone needing a review in some basic algebra and trig topics. The review contains the occasional
More informationGrades 7 & 8, Math Circles 10/11/12 October, Series & Polygonal Numbers
Faculty of Mathematics Waterloo, Ontario N2L G Centre for Education in Mathematics and Computing Introduction Grades 7 & 8, Math Circles 0//2 October, 207 Series & Polygonal Numbers Mathematicians are
More informationInstructor (Brad Osgood)
TheFourierTransformAndItsApplications-Lecture26 Instructor (Brad Osgood): Relax, but no, no, no, the TV is on. It's time to hit the road. Time to rock and roll. We're going to now turn to our last topic
More informationMath 308 Midterm Answers and Comments July 18, Part A. Short answer questions
Math 308 Midterm Answers and Comments July 18, 2011 Part A. Short answer questions (1) Compute the determinant of the matrix a 3 3 1 1 2. 1 a 3 The determinant is 2a 2 12. Comments: Everyone seemed to
More informationUniversity of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra
University of Colorado at Colorado Springs Math 090 Fundamentals of College Algebra Table of Contents Chapter The Algebra of Polynomials Chapter Factoring 7 Chapter 3 Fractions Chapter 4 Eponents and Radicals
More informationMITOCW ocw-18_02-f07-lec02_220k
MITOCW ocw-18_02-f07-lec02_220k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.
More informationMITOCW ocw f99-lec16_300k
MITOCW ocw-18.06-f99-lec16_300k OK. Here's lecture sixteen and if you remember I ended up the last lecture with this formula for what I called a projection matrix. And maybe I could just recap for a minute
More informationPre-calculus is the stepping stone for Calculus. It s the final hurdle after all those years of
Chapter 1 Beginning at the Very Beginning: Pre-Pre-Calculus In This Chapter Brushing up on order of operations Solving equalities Graphing equalities and inequalities Finding distance, midpoint, and slope
More informationMITOCW ocw f99-lec23_300k
MITOCW ocw-18.06-f99-lec23_300k -- and lift-off on differential equations. So, this section is about how to solve a system of first order, first derivative, constant coefficient linear equations. And if
More informationTo Find the Product of Monomials. ax m bx n abx m n. Let s look at an example in which we multiply two monomials. (3x 2 y)(2x 3 y 5 )
5.4 E x a m p l e 1 362SECTION 5.4 OBJECTIVES 1. Find the product of a monomial and a polynomial 2. Find the product of two polynomials 3. Square a polynomial 4. Find the product of two binomials that
More informationMITOCW ocw f07-lec39_300k
MITOCW ocw-18-01-f07-lec39_300k The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources for free.
More informationSolving Quadratic & Higher Degree Inequalities
Ch. 10 Solving Quadratic & Higher Degree Inequalities We solve quadratic and higher degree inequalities very much like we solve quadratic and higher degree equations. One method we often use to solve quadratic
More informationEgyptian Fractions: Part I
Egyptian Fractions: Part I Prepared by: Eli Jaffe October 8, 2017 1 Cutting Cakes 1. Imagine you are a teacher. Your class of 10 students is on a field trip to the bakery. At the end of the tour, the baker
More informationAlgebra 2 Summer Work Packet Review and Study Guide
Algebra Summer Work Packet Review and Study Guide This study guide is designed to accompany the Algebra Summer Work Packet. Its purpose is to offer a review of the nine specific concepts covered in the
More informationError Correcting Codes Prof. Dr. P Vijay Kumar Department of Electrical Communication Engineering Indian Institute of Science, Bangalore
(Refer Slide Time: 00:54) Error Correcting Codes Prof. Dr. P Vijay Kumar Department of Electrical Communication Engineering Indian Institute of Science, Bangalore Lecture No. # 05 Cosets, Rings & Fields
More information32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE
get the complete book: /getfulltextfullbook.htm 32. SOLVING LINEAR EQUATIONS IN ONE VARIABLE classifying families of sentences In mathematics, it is common to group together sentences of the same type
More informationThe PROMYS Math Circle Problem of the Week #3 February 3, 2017
The PROMYS Math Circle Problem of the Week #3 February 3, 2017 You can use rods of positive integer lengths to build trains that all have a common length. For instance, a train of length 12 is a row of
More informationPROFESSOR: WELCOME BACK TO THE LAST LECTURE OF THE SEMESTER. PLANNING TO DO TODAY WAS FINISH THE BOOK. FINISH SECTION 6.5
1 MATH 16A LECTURE. DECEMBER 9, 2008. PROFESSOR: WELCOME BACK TO THE LAST LECTURE OF THE SEMESTER. I HOPE YOU ALL WILL MISS IT AS MUCH AS I DO. SO WHAT I WAS PLANNING TO DO TODAY WAS FINISH THE BOOK. FINISH
More informationLesson 21 Not So Dramatic Quadratics
STUDENT MANUAL ALGEBRA II / LESSON 21 Lesson 21 Not So Dramatic Quadratics Quadratic equations are probably one of the most popular types of equations that you ll see in algebra. A quadratic equation has
More informationMITOCW MITRES18_005S10_DiffEqnsMotion_300k_512kb-mp4
MITOCW MITRES18_005S10_DiffEqnsMotion_300k_512kb-mp4 PROFESSOR: OK, this lecture, this day, is differential equations day. I just feel even though these are not on the BC exams, that we've got everything
More informationFog Chamber Testing the Label: Photo of Fog. Joshua Gutwill 10/29/1999
Fog Chamber Testing the Label: Photo of Fog Joshua Gutwill 10/29/1999 Keywords: < > formative heat&temp exhibit interview 1 Goals/Context Fog Chamber Interview Results Testing Label: Photo of fog on Golden
More informationRational Expressions and Radicals
Rational Expressions and Radicals Rules of Exponents The rules for exponents are the same as what you saw in Section 5.1. Memorize these rules if you haven t already done so. x 0 1 if x 0 0 0 is indeterminant
More informationLesson 3: Advanced Factoring Strategies for Quadratic Expressions
Advanced Factoring Strategies for Quadratic Expressions Student Outcomes Students develop strategies for factoring quadratic expressions that are not easily factorable, making use of the structure of the
More informationAlgebra II. Slide 1 / 276. Slide 2 / 276. Slide 3 / 276. Polynomials: Operations and Functions. Table of Contents
Slide 1 / 276 lgebra II Slide 2 / 276 Polynomials: Operations and Functions 2014-10-22 www.njctl.org Table of ontents click on the topic to go to that section Slide 3 / 276 Properties of Exponents Review
More informationRolle s Theorem. The theorem states that if f (a) = f (b), then there is at least one number c between a and b at which f ' (c) = 0.
Rolle s Theorem Rolle's Theorem guarantees that there will be at least one extreme value in the interior of a closed interval, given that certain conditions are satisfied. As with most of the theorems
More informationMath 58 Test # 3 Fall 2014 Chapter 9, Part 1 Instructor: Smith-Subbarao
Math 58 Test # 3 Fall 2014 Chapter 9, Part 1 Instructor: Smith-Subbarao Solutions Name Score: If you are not sure about notation, please ask for help! (4 + 1 ) / x means 4 + 1 is in numerator and x is
More informationACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER /2018
ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 2017/2018 DR. ANTHONY BROWN 1. Arithmetic and Algebra 1.1. Arithmetic of Numbers. While we have calculators and computers
More informationIn this unit we will study exponents, mathematical operations on polynomials, and factoring.
GRADE 0 MATH CLASS NOTES UNIT E ALGEBRA In this unit we will study eponents, mathematical operations on polynomials, and factoring. Much of this will be an etension of your studies from Math 0F. This unit
More informationUnit 1 - Algebra. Chapter 3 Polynomials Chapter 4 Equations MPM1D
Unit 1 - Algebra Chapter 3 Polynomials Chapter 4 Equations MPM1D Chapter 3 Outline Section Subject Homework Notes Lesson and Homework Complete (initial) 3.2 Work With Exponents 3.3a Exponent Laws 3.3b
More informationCAHSEE on Target UC Davis, School and University Partnerships
UC Davis, School and University Partnerships CAHSEE on Target Mathematics Curriculum Published by The University of California, Davis, School/University Partnerships Program 2006 Director Sarah R. Martinez,
More informationAlgebra I. Polynomials.
1 Algebra I Polynomials 2015 11 02 www.njctl.org 2 Table of Contents Definitions of Monomials, Polynomials and Degrees Adding and Subtracting Polynomials Multiplying a Polynomial by a Monomial Multiplying
More informationEgyptian Fractions: Part I
Egyptian Fractions: Part I Prepared by: Eli Jaffe October 8, 2017 1 Cutting Cakes 1. Imagine you are a teacher. Your class of 10 students is on a field trip to the bakery. At the end of the tour, the baker
More informationVolume vs. Diameter. Teacher Lab Discussion. Overview. Picture, Data Table, and Graph
5 6 7 Middle olume Length/olume vs. Diameter, Investigation page 1 of olume vs. Diameter Teacher Lab Discussion Overview Figure 1 In this experiment we investigate the relationship between the diameter
More informationSolving Equations by Adding and Subtracting
SECTION 2.1 Solving Equations by Adding and Subtracting 2.1 OBJECTIVES 1. Determine whether a given number is a solution for an equation 2. Use the addition property to solve equations 3. Determine whether
More informationMITOCW ocw nov2005-pt1-220k_512kb.mp4
MITOCW ocw-3.60-03nov2005-pt1-220k_512kb.mp4 PROFESSOR: All right, I would like to then get back to a discussion of some of the basic relations that we have been discussing. We didn't get terribly far,
More informationACCUPLACER MATH 0310
The University of Teas at El Paso Tutoring and Learning Center ACCUPLACER MATH 00 http://www.academics.utep.edu/tlc MATH 00 Page Linear Equations Linear Equations Eercises 5 Linear Equations Answer to
More informationChapter 6, Factoring from Beginning and Intermediate Algebra by Tyler Wallace is available under a Creative Commons Attribution 3.0 Unported license.
Chapter 6, Factoring from Beginning and Intermediate Algebra by Tyler Wallace is available under a Creative Commons Attribution 3.0 Unported license. 2010. 6.1 Factoring - Greatest Common Factor Objective:
More informationCOLLEGE ALGEBRA. Paul Dawkins
COLLEGE ALGEBRA Paul Dawkins Table of Contents Preface... iii Outline... iv Preliminaries... 7 Introduction... 7 Integer Exponents... 8 Rational Exponents...5 Radicals... Polynomials...30 Factoring Polynomials...36
More informationModern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur
Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture 02 Groups: Subgroups and homomorphism (Refer Slide Time: 00:13) We looked
More informationEQ: How do I convert between standard form and scientific notation?
EQ: How do I convert between standard form and scientific notation? HW: Practice Sheet Bellwork: Simplify each expression 1. (5x 3 ) 4 2. 5(x 3 ) 4 3. 5(x 3 ) 4 20x 8 Simplify and leave in standard form
More informationMITOCW 2. Harmonic Oscillators with Damping
MITOCW 2. Harmonic Oscillators with Damping The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources
More informationMITOCW MIT8_01F16_w02s05v06_360p
MITOCW MIT8_01F16_w02s05v06_360p One of our classic problems to analyze using Newton's second law is the motion of two blocks with a rope that's wrapped around a pulley. So imagine we have a pulley, P,
More informationJune If you want, you may scan your assignment and convert it to a.pdf file and it to me.
Summer Assignment Pre-Calculus Honors June 2016 Dear Student: This assignment is a mandatory part of the Pre-Calculus Honors course. Students who do not complete the assignment will be placed in the regular
More information