Binomial Expansions-Math Reflection By: Hannah Gemei 8D
Introduction Math is the language of science and engineering, it is the tool to solve problems. Math language and engineering can :A binomial is a mathematical expression of two unlike terms with coefficients and which is raised to at least the power of 1. Examples of binomials are (a+b), (x+2)^2, (a+c)^3, (y-4)^6.
If you were an engineer 100 years ago, explain how our method may have been useful rather than just using long multiplication? If I was an engineer 100 years ago, the formula (a+b)^2 =a^2 +2ab+b^2 and (a-b)^2 =a^2 -2ab+b^2 , would save time, long ago, and also help solve problems such as: To calculate the labor rates and percentage of sick days. Traditional results analysis methods that can provide an intuitive approach to valuing projects with flexible management, or real options. 100 years ago, engineers could have used a binomial conclusion tree with risk-neutral probabilities to evaluate what will happen t the project overtime. Binomials can also be used to model different situations, like in the stock market of engineering companies to see how the prices will vary over time. If the engineers had this tool100 years ago they would have achieved much more successful projects.
Example, (2) Since polynomials (special kind of binomials) are used to describe curves of various types, people use them in the real world to graph curves. For example, roller coaster designers may use polynomials to describe the curves in their rides. Combinations of polynomial functions are sometimes used in economics to do cost analyses. 100 years ago the engineers could make the families happier with these, roller coasters and fun fairs.
Example (3) Another example if I was an engineer 100 years ago, I will use the formula to design, water canals, the binomial expansion will help me calculate the slope, cross sectional area and flow rates of the water. Also the formula will help me engineer roads, by calculating the volume of gravel and asphalt and multiply the quantity needed for the whole road.
At What point would our method be big and cumbersome? (ie. Hw many decimal places or what sorts of numbers would make you think twice about using this method?) A binomial is an algebraic expression containing 2 terms. We sometimes need to expand binomials as follows: • (a + b)0 = 1 • (a + b)1 = a + b • (a + b)2 = a2 + 2ab + b2 • (a + b)3 = a3 + 3a2b + 3ab2+b3 • (a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3+b4 • (a + b)5 = a5 + 5a4b + 10a3b2 + 10a2b3 + 5ab4+b5 This Can be rather difficult for larger powers or more complicated expressions. And numbers with lots of decimals.
Alternative methods • The coefficients (the numbers in front of each term) follow a pattern. • 1 1 • 1 2 1 • 1 3 3 1 • 1 4 6 4 1 • 1 5 10 10 5 1 • 1 6 15 20 15 6 1 • You can use this pattern to form the coefficients, rather than multiply everything out as we did above. • Our method is good to use if a company is using certain numbers repeatedly, in her/his data system, but if they have numbers exceeding 170, it will not be a practical formula.
Can you give us some detailed examples and explanations of where long multiplication is more efficient than our expansion method? The binomial theorem describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the power (x + y)n into a sum involving terms of the form axbyc, where the coefficient of each term is a positive integer, and the sum of the exponents of x and y in each term is n. For example, the coefficients appearing in the binomial expansion are known as binomial coefficients. They are the same as the entries of Pascal's triangle( the one in the 7th slide). The most basic example of the binomial theorem is the formula for square of x + y: (x+y)^2=x^2+2xy+y^2. But for long multiplication, it is a special method for multiplying larger numbers.It is a way to multiply numbers larger than 10 that only needs your knowledge of the ten times Multiplication Table. In that case it is easier to use the long multiplication. Let us say we want to multiply, 612 × 24: • First we multiply 612 × 4 (=2,448), • then we multiply 612 × 20 (=12,240), • and last we add them together (2,448+12,240=14,688).
Development of Long Multiplication Another example of long multiplication, is this diagonal grid that was published in Chicago Tribune, a lot of long multiplication grids, are being invented and renewed to make the calculations shorter, and time consuming. An example of using long multiplication is Genetic Engineering: The graph shows one use of long multiplication for DNA sequencing applications, lab results. Completely sequenced genomes in Genomes On-Line Database (GOLD) and Integrated Microbial Genomes (IMG)
Development of Long Multiplication Another application for long multiplication is Weather forecasting, which is an application for science technology and math, it tells the current and future of weather and sky conditions. It also impacts the agriculture, and trade markets. The nature of the atmosphere makes it to requires massive power to solve the equations to solving the atmosphere, these equations have to be very accurate since they impact life and property. The simple long multiplications is part of these equations. Conclusion: These examples prove that the long multiplication can sometimes be more efficient than our binomial method. In Genetic Engineering, it can change the face of the future, in the Weather Forecast, the right calculations can save lives and property. The data and and calculations helped the scientists predict the Rita hurricane in 2005.