Math117 Discussion Questions
Math117 Discussion Questions
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Date Created: 11/16/15
Week 1 DQ 1 Quotient Rule: This rule states when a base number, raised to an exponent, is divided by the same base number, which is raised to a different exponent, the result equals the base number raised by the first exponent minus the second exponent. This rule states that the base number cannot equal zero. Product Rule: This rule states when a base number, raised to an exponent, is multiplied by the same base number, raised to a different exponent, the result equals the base number raised by the first exponent plus the second exponent. Power Rule: This rule states when a base number, raised to an exponent, which is then raised by another exponent, the result equals the base number raised by the first exponent multiplied by the second. Solve the following: 3.2 x 20^2= The three rules for exponents listed in the chart on page 239 of this week’s text include: the product rule, the quotient rule, and the power rule. The Product Rule states that when the a base number raised to an exponent is multiplied by the same base number raised to an different exponent, the answer equals the base number raised to the first exponent plus the second exponent. For example: (52)(53) = 52+3 = 55 The Quotient Rule states that when the a base number raised to an exponent is divided by the same base number raised to an different exponent, the answer equals the base number raised to the first exponent minus the second exponent. With this rule, the base number cannot equal zero. For example: (55)/(53) = 553 = 52 The Power Rule states that when a base number raised to an exponent is raised to an exponent, the answer equals the base number raised to the first exponent multiplied by the second exponent. For example: (55)2 = 55 x 2 = 510 Solve the following: 2.4 x (104)2 = Week 1 DQ 2 Dividing a polynomial by a binomial is similar to the long division learned in elementary school in the sense that the same steps are taken to solve the problem. To carry out long division, you must first divide, then multiply, then subtract, and then bring down the next term. This rule still holds true using binomials and polynomials. Understanding how one kind of division works definitely helps understand the other kind because the fundamentals are the same. It does not matter if you are using numbers, letters, exponents, or a combination of them all, if you understand the fundamentals of long division you can solve any division problem. An example of how polynomial division is used in real life is by architects. Architects use polynomial division when designing buildings and their layouts. I can not think of too many other professionals besides math majors/students who would use polynomial division in real life. When dividing a binomial or polynomial, you must first write out the problem in long division. The basic operations are still followed, but with a binomial or polynomial, you are dividing two or more numbers at a time. What I mean is that you divide the first number into the first number. Then you multiply the result to the second number. These first two results are subtracted from the first two numbers that are being divided. Then you bring down the next term and repeat until the degree of the remainder is less than zero. When the quotient is expressed, the remainder is shown in fraction form over the divisor. Don’t forget to fill in the missing terms, usually expressed as x³ + 0x² + x. I work in a manufacturing plant, and I see different types of algebra being used each day. It takes a certain number of fasteners (f), wires (w), one evaporator (e), one condenser (c), and two compressors (cp²) to build a unit. If it takes 100 fasteners and 24 wires, one evaporator, one condenser, and two compressors to build a unit, we can use the expression 100f + 24w + e + c + cp² = u There are 10,000 fasteners in stock, 240 wires in stock, 30 evaporators, 30 condensers, and 90 compressors in stock. How many units can be built? Week 3 DQ 1 1. Take any number (except 1). Square that number and then subtract one. Divide by one less than your original number. Now subtract your original number. Did you reach 1 for an answer? You should have . 7: 7 1 = 48 48/ 6 = 8 8 – 7 = 1 2. How does this number game work? (Hint: redo the number game using a variable instead of an actual number and rewrite the problem as one rational expression). x: (x – 1)/(x – 1) = (x + 1)(x – 1)/(x – 1) = (x + 1) (x + 1) – x = 1 3. How did the number game use the skill of simplifying rational expressions? Once you use a variable in the number game vise a number, it is much easier to see how the number game used the skill of simplifying rational expressions to get the same answer each time. Using a variable allowed us to cancel out an expression from both the numerator and the denominator, leaving a single expression. From there, it was simple math that brought us to the answer “1.” 4. Create your own number game using the rules of algebra and post it for your classmates to solve. Be sure to think about values that may not work. State whether your number game uses the skill of simplifying rational expressions Pick any whole number. Now double it. Add ten to your answer. Then divide your answer by two. Subtract your original number from the end result. Your answer should be 5. This game uses the skill of simplifying rational expressions. Take any number (except for 1). Square that number and then subtract one. Divide by one less than your original number. Now subtract your original number. You reached 1 for an answer, didn’t you? How does this number game work? (Hint: Redo the number game using a variable instead of an actual number and rewrite the problem as one rational expression). How did the number game use the skill of simplifying rational expressions? Create your own number game using the rules of algebra and post it for your classmates to solve. Be sure to think about values that may not work. State whether your number game uses the skill of simplifying rational expressions. 2 2 =4 3/1=3 32=1 x2 2 x 1 x 1/x1 ((x 1)/(x1))x (x+1)(x1)/(x1)x x+1x =1 Because you have to factor and simplify like terms. Number game: Take any number, square it, and add 6 multiplied by the original number. After that add 5. Now divide this by the original number plus one. Now subtract your original number to get 5 Week 3 DQ 2 1. How is doing operations (adding, subtracting, multiplying, and dividing) with rational expressions similar to or different from doing operations with fractions? Doing operations with rational expressions is similar to doing operations with fractions in the sense that all the rules are the same for both. For example, when subtracting or adding rational expressions, the denominators must be the same in order to complete the equation. This is the same case for fractions. In order to make the denominators the same in either equation, you must identify the LCD (Least Common Denominator) and multiply the fraction or rational expression by whatever is needed to make the denominator equal to the LCD. In addition to the rules for adding and subtracting, the same rules apply to both fractions and rational expressions when multiplying and dividing. When multiplying either ration expressions or fractions, you simply multiply the numerators then multiply the denominators, and then simplify. When dividing either rational expressions or fractions you simply multiply the first rational expression or fraction by the reciprocal of the second rational expression or fraction. 2. Can understanding how to work with one kind of problem help understand how to work another type? If you understand how to work one kind of problem, either fractions or rational expressions, you will most certainly understand how to work the other. The fundamentals are the same for both. The only difference is that rational expressions may require a little more work and understanding algebra as compared to regular fractions. 3. When might you use this skill in real life? You might use the skill doing operations with rational numbers in real life for any number of reasons. Doing operations with rational expressions is similar to doing operations with fractions. Subtracting or adding rational expressions requires the denominators to be the same. The same is true when evaluating fractions. In order for the denominators to be the same in either equation, you must identify the least common denominator (LCD) and multiply the fraction, or rational expression, by whatever number that is required to make the denominators equal. The same rules apply to both fractions and rational expressions when multiplying and dividing. You must multiply the numerators, then multiply the denominators, and then simplify. The fundamentals when working with either fractions or rational expressions are the same. Understanding how to use these operations correctly with any kind of problem will help you understand how to better use these same operations in other equations or problems. This can be used in real life when doing basic tasks such as shopping or budgeting. Week 5 DQ 1 Why is it important to simplify radical expressions before adding or subtraction? It is important to simplify radical expressions before adding or subtraction in order to get like radical terms to equate. Once you collect like radical terms, you may then add or subtract as required. For example you cannot add the radical expressions 4√8 and 3√2 as is. To solve this equation, you must simplifying the radical expressions first and then add as followed: 4√8 + 3√2 = 4(√4 * 2) + 3√2 = 4 * 2 * √2 + 3 √2 = 8√2 + 3√2 = 11√2 How is adding radical expression similar to adding polynomial expressions? Adding radical expressions is similar to adding polynomial expressions for the simple fact that in order to add either one, you must first have like terms. Trying to add √8 and 2 √2 in radical expressions is similar to trying to add x and 3x with polynomial expressions, it cannot be done because they are not like terms. Adding both radical expressions and polynomials are also similar in the fact that you can use the distributive property to solve them. For example: 4x + 5x = (4 + 5) x = 9x 2 5√x + √x = (5+1)√x = 6√x How is it different? Adding radical expressions is different from adding polynomial expressions Provide a radical expression for your classmates to simplify. Solve the following: 5 √192x + 2 √3x Answer… 5 √192x + 2 √3x = 3 3 3 5 √64x * 3x + 2 √3x = 3 3 3 3 5 √64x + √ 3x + 2 √3x = 3 3 5 * 4x * √ 3x + 2 √3x = 3 3 20x √ 3x + 2 √3x = 3 (20x + 2) √3x It is important to simplify all expressions before adding or subtracting. Therefore it should be no different for radical expressions! In any math problem you encounter it is always a rule of thumb to simplify as much as you possibly can before trying to solve an equation. Using the example of PEDMAS, it takes the form of exponents. Radical expressions can be treated the same in the order in which you solve an equation. If an equation was not simplified the solution would not be solved completely and the answer would not be correct. Solve: 2(√5 +√2) / 3(√2 √5) Week 5 DQ 2 1. Review section 10.2 (p,692) of your text. Describe two laws of exponents. Of the five laws of exponents listed in this week’s text, I chose to describe the law when multiplying a power by a power and when raising a power to a power. Multiplying a power by a power: When multiplying a power by a power when the base numbers are the same, you simply keep the base number and add the exponents. Raising a power to a power: When raising a power to a power you keep the base the same and multiply the exponents. 2. Provide an example illustrating each law. Multiplying a power by a power: x2 * x5 = x7 Raising a power to a power: (x2)5 = x10 3. Explain how to simplify your expression ? Multiplying a power by a power: When multiplying a power by a power when the base numbers are the same, you simply keep the base number and add the exponents. Raising a power to a power: When raising a power to a power you keep the base the same and multiply the exponents. 4. How do the laws work with rational exponents? These laws of exponents work the same with rational exponents as they do with regular exponents. The fundamentals still remain the same. 5. Provide the class with a third expression to simplify that includes rational (fractional) exponents. Solve the following… (x2/3 * x1/9)/ (x5/6 * x1/3) = First rule is Multiplication of exponents: When multiplying exponents you keep the base the same and add the exponents together: x^3(x^6) = x^9 When dividing exponents you also keep the base the same and this time subtract the denominator's exponent from the numerator's exponent: x^7 x^8 = x^-1 The laws to exponents do not change when using radical expressions. Example: y^7(y^6) y^5(y^8) Week 7 Dq 1 How do you know if a quadratic equation will have one, two, or no solutions? You can know that the quadratic equation will have one solution if you compute the discriminant first. If it is 0 and only one intercept it is a real number therefore there will only be one solution. If it ends up a positive, there will be two solutions. And if it is a negative, we will be taking the square root of a negative number, making two nonreal complex number solutions and they will be complex conjugates or no solution. How do you find a quadratic equation if you are only given the solution? To solve the answer to a quadratic equation when you have only the solution you must use the principle of zero in rev. This means you must set it to zero (0), then clear out any fractions or anything in the way and get all the coefficients on one side. Then using the reverse principle of zero and finishing using the FOIL method and collect like terms if needed. . Is it possible to have different quadratic equations with the same solution? Explain. Yes it is possible to have different quadratic equations with the same solution. In order for this to happen you must insert an extra letter for clarification. You then factor and use the principle of zero and re substitute the original letter for the one you substituted. Then when the power is raised you will have the same solution. Provide your classmates with one or two solutions with which they must create a quadratic equation. Try this: The solutions are 2+ ^2 and 2 ^2. D = b^2 - 4c: When D > 0, P(x) has two distinct real roots. When D = 0, P(x) has two coincident real roots. When D < 0, P(x) has no real roots. How do you find a quadratic equation if you are only given the solution? roots x = a and x = -b, then the quadratic equation is (x-a)(x+b) = 0. Is it possible to have different quadratic equations with the same solution? Yes, it is! Again, if we have the roots x = a and x = -b, then we can have C(x-a)(x+b) = 0, where C is a constant. So multiplying with a constant does not change the roots. Provide your classmate's with one or two solutions with which they must create a quadratic equation. Find a quadratic equation that has roots x = 2 and x = 5: (x-2)(x-5) = 0. Find a quadratic equation that has two roots at x = 1: (x-1)(x-1) = 0. You can determine how many, if any, solutions a quadratic equation may have by looking at the discriminant. If the discriminant is greater than 0, then there will be two distinct real roots. If the discriminant is equal to 0, then it will have two coincident real roots. And, if the discriminant is less that 0, it will no real root. When only provided with the solution, you must utilize the zero products principal in reverse, as well as using the FOIL method to collect like terms, in order to find the quadratic equation. It is possible to have the same solution for a quadratic equation when multiplying with constant. By doing so, it doesn’t change the roots. Find a quadratic equation that has roots x = 3 and x = 4: (x3)(x4) = 0. Find a quadratic equation that has two roots at x = 2: (x2)(x2) = 0. Week 7 DQ 2 Quadratic equations can be solved by graphing, using the quadratic formula, completing the square, and factoring. What are the pros and cons of each of these methods? When might each method be most appropriate? Which method do you prefer? Why? Instructor and Classmates, A. Factoring: Factoring in most cases can solve Quadratic equations. The pros to using factoring are quite simple; it is usually the easier method to use but not always. It’s also pretty accurate. However there are times where factoring cannot be used. The cons to factoring can also be when factoring some types of problems containing Polynomials. However, when factoring it can be broken down into 5 easy steps, because of these steps I prefer factoring. Step1. You write the equation in standard form: ax^2 + b x + c = 0. Step2. Factor the quadratic completely. Step3. Set each factor containing a variable equal to 0. Step4. Solve the resulting equations. Step5. Check each solution in the original equation. I think the Factoring Method is the easier method for most people to use when working on Quadratic equations. B. Completing the square: This method is also a great way to do quadratic equations if you already have certain information about the equation. One of the Cons to the Completing the Square Method is it can be frustrating and overwhelming at times when dealing with coefficients and knowing when to divide. This does compute by using five steps however, they can be more complicated than factoring. For example, Step1. If the coefficient of x^2 is 1, go to step 2. Otherwise, divide both sides of the equation by the coefficient of x^2. Then Step2. Isolate all variable terms on one side of the equation, this is easier said than done sometimes. Step3. Complete the square for resulting binomials by adding the square of half of the coefficient of x to both sides of the equations. A lot of people forget to do this step and get the incorrect answers. Step4. Factor the resulting perfect square trinomial and write it as a square of the binomial. Step5. Use the square root property to solve for x. Therefore, completing the square in my opinion makes solving the quadratic equation much more complicated and difficult. It is my least favorite method because of all the confusing steps needed to solve it properly. As some other students feel, this may be a quicker method when the problem is a simple one however it can lead to many errors if the problem has multiple parts or is more than just a small equation. C. The Quadratic Formula is used to solve any equation written in standard form ax^2 + b x +c = 0 as long as (a) is not a zero. If you know the discriminate you can determine the number and type of solutions of each quadratic equation. This would be a Pro at minimum. The quadratic formula is very useful when solving problems modeled by the quadratic equations. Obviously, if one can determine solutions or equations that are modeled in this way, this would be the best method to use when solving this equation. I also like using this method because it can be done or used in four simple steps. Step1. If the equation is in the form (ax + b)^2, you use the square root property and solve. If not, you go to step 2. Step2. Write the equation in the standard form: ax^2 + b x + c = 0. Step3. You will try to solve the equation by factoring method. If that is not possible, go to Step 4. Which is, solve the equation by the quadratic formula. This can be used to solve radical equations and others as well. The steps here are almost like trouble shooting in some ways, establishing exactly which way you are going to finally solve it. The downside or Con to this method is it is a lot harder when you are missing certain information needed to solve the equation. I prefer to use factoring a lot bet er. D. Graphing Method: Although this method can actually end up being the most complex way of working a quadratic equation, it is easier for me because I am a more visual learner. By using the graphing methods we are expose to many different formulas depending which way our plots or intersects are pointing on the graph. Of course it may be faster if our plots are already known and marked, however, this tends to be the most inaccurate method at times. If you have a certain amount of information already it may be more accurate. You must however be able to find out the vertex of the parabola and know when it is facing upward or downward. For a lot of students this method can be confusing and overwhelming and not as accurate as one would prefer their answers to be. Again, I believe the Pro to be that it brings a more visual aspect to the working equation itself, which I personally and a lot of other students benefit from seeing rather than just reading it. The con with graphing can be many. Less accurate and more complex depending on the information already in use on the graph. This is one method that only gets easier with practice. E. Overall, I would finally say that the factoring method is more than likely the most comfortable way of solving when possible. I would then use graphing if I had enough information and the answer just needed to be in the ballpark. All four of these methods have pros and cons to them. Some work better than others during certain problems. And we can use all of them better with practice. What I think is really neat is the fact that we can actually solve the same equation and get the same answer by using a different method. Does everyone else agree? I think that most of us do agree that we all think differently and some of us feel better using some methods more often than others. Appreciate the feedback. Graphing can help picture the equation, however, is not an exact method for solving quadratic equations. The quadratic formula is the preferred method when solving applied problems because it is faster. The quadratic formula can also be useful since the solutions of quadratic equations cannot always be found by factoring, but can always be found using the quadratic formula. Completing the square can be difficult since you have many steps to remember and carry out in order to solve quadratic equations. Completing the square is useful for providing a base for proving the quadratic formula. Factoring can be easy to use for quadratic equations that are in a simple form, but for more complex quadratic equations may be more difficult to use. In order to decide when each method might be most appropriate you would have to think about your goal. If you want a general idea or a to be able to picture the solution you might want to use the graphing method. If you want an absolute solution with little steps you might want to use the quadratic formula or if the amount of steps is not important you could choose from either the quadratic formula or completing the square. If the quadratic equation is in a simple form you may want to use factoring, especially if the answer seems obvious. As a general rule I would probably choose to solve quadratic equations using the quadratic formula since it can always be used to find the solutions and it has fewer steps than completing the square. Graphing is not an exact method for solving quadratic equations. The quadratic formula is a faster and preferred method for solving problems. Factoring, however, doesn’t always produce a solution for quadratic equations. Quadratic equations in simple form may be easier to solve utilizing factoring; but, more complex quadratic equations are much more difficult to solve utilizing this method. There are many steps to remember when completing the square in order to solve quadratic equations. However, completing the square provides a base for proving the quadratic formula. The desired response will dictate the best method to utilize. General, but not exact, solutions would use the graphing method. Specific solutions would require the use of the quadratic formula or factoring. Week 9 Capstone DQ I consider the content of this course, and math in general, to be a useful tool. Not only do I enjoy math but I also find that I use it on a daily basis, from budgeting my finances to clipping coupons and adding up my savings when shopping. As far as the concepts from this course, they too can be applied to my life. I never thought that many of these mathematical concepts could be applied to everyday life, such as rational equations, radical equations, Pythagorean Theorem, and proportions. These types of concepts could be used to determine such things as: a) How much time I could save if someone was helping me do a certain job ? b) How far I can see into the distance from some peak ? c) What the wind chill temperature is outside ? So I can plan accordingly. I've taken full advantage of the Center for Mathematic Excellence which has assisted me in developing my mathematics skills on several occasions. I have also found the MyMathLab to be a useful study tool as well. I think it was a great idea for Axia of UOP to list information pertaining to Math Anxiety. This information can allow students to understand what is it, how it works, and what you can do to develop strategies that will eliminate problems. Once you have developed math strategies, you are bound to become an excellent student who will then succeed in class. What I liked is that it explained in detail how to do CheckPoint problems or where to find information on problems in the text so you did not have to spend a lot of time searching for things. With concepts, you may have needed extra work on before a test, or with so many things to remember from the text, you could easily refresh your memory with the conveniently accessible study plans. I think between the instructors, the Center for Mathematic Excellence, and MyMathLab, the University of Phoenix has done a good job to make it as easy as possible for students to succeed in their math courses. MAT/117 has reinforced the idea of math being a useful tool. I work at a manufacturing plant that assembles commercial water chillers. There are many types and quantities of parts used in building one unit. The overall time from placement of the order to completion and delivery also varies. There are several applications that I have learned over the last eight weeks that are used on a daily basis. For example, our sales office deals with the type of equation mention in appendix f when forecasting our production schedule. The parts department also uses various polynomials equations when deciding what to purchase for inventory. On a personal note, I occasionally undertake a carpentry project, which utilizes the slope of a line. I now have a better understanding of how to determine the rise and the run of my projects. I can also determine the grade of an incline while working on my model railroad. The primary use of MyMathLab was to use the explanation tab when I got stuck or needed help with a problem. I also used the animations to try and better understand certain concepts. The videos, animations, and Power Point slide shows in the course material were also useful tools used during this course. Best of luck to you all! The content in this course has definitely allowed me to think of math as a useful tool. I know I learned the Pythagorean Theorem years ago when I was in high school and forgot how useful that one concept could be. I use basic math in my personal and professional life for everything such as counting change to adjusting recipes to fit my needs. I used MyMathLab constantly especially during my checkpoints to see how each step of a problem is worked. I took a lot of notes and seemed to do well until it was time for a quiz. This class was a reality check for me. I thought I was fairly good at math, but this class opened my eyes to how much I did not know. I learned a lot and have realized that I still have a lot to learn.
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