Mathematics
Integration Using Long Division
Integration using long division is a method of integrating rational functions that involves dividing the numerator by the denominator using long division. This process allows the rational function to be expressed as a sum of simpler functions that can be integrated more easily. The method is particularly useful for integrating functions with quadratic denominators.
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3 Key excerpts on "Integration Using Long Division"
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
For example, if you were asked to divide 6542 by 21, the long division method used is illustrated in the margin. This solution can be written two ways: 311 R11 or 311 + 11 ___ 21 . In this example, the dividend is 6542, the divisor is 21, the quotient is 311, and the remainder is 11. We employ a similar technique (dividing the leading terms) when dividing polynomials. 4.3.1 Long Division of Polynomials 4.3.1 Skill Divide polynomials with long division. 4.3.1 Conceptual Extend long division of real numbers to polynomials. Let’s start with an example whose answer we already know. We know that a quadratic expression can be factored into the product of two linear factors: x 2 + 4x − 5 = (x + 5)(x − 1). Therefore, if we divide both sides of the equation by (x − 1), we get x 2 + 4x − 5 __________ x − 1 = x + 5 We can state this by saying x 2 + 4x − 5 divided by x − 1 is equal to x + 5. Confirm this statement by long division: x − 1 x 2 + 4x − 5 __ ⟌ 6542 _ 21⟌ 311 − 63 ‾ 24 − 21 ‾ 32 − 21 ‾ 11 For each polynomial in Exercises 107 and 108, determine the power function that has similar end behavior. Plot this power function and the polynomial. Do they have similar end behavior? 107. f (x) = −2x 5 − 5x 4 − 3x 3 108. f (x) = x 4 − 6x 2 + 9 In Exercises 109 and 110, use a graphing calculator or a computer to graph each polynomial. From the graph, estimate the x-intercepts and state the zeros of the function and their multiplicities. 109. f (x) = x 4 − 15.9x 3 + 1.31x 2 + 292.905x + 445.7025 110. f (x) = −x 5 + 2.2x 4 + 18.49x 3 − 29.878x 2 − 76.5x + 100.8 In Exercises 111 and 112, use a graphing calculator or a computer to graph each polynomial. From the graph, estimate the coordinates of the relative maximum and minimum points. Round your answers to two decimal places. 111. f (x) = 2 x 4 + 5 x 3 − 10 x 2 − 15x + 8 112. f (x) = 2 x 5 − 4 x 4 − 12 x 3 + 18 x 2 + 16x − 7
eBook - ePub- Gaea Leinhardt, Ralph Putnam, Rosemary A. Hattrup(Authors)
- 2020(Publication Date)
- Routledge(Publisher)
Real Math series, but there is some mathematical justification given for the procedure. The "long division algorithm" is not introduced in this series until fifth grade, and it is not supposed to be taught until students have done considerable work on rates, ratio, equal proportions, decimal numbers, and functions.In relation to the problem of dividing 414 by 24, the following directions are given:(Willoughby, Bereiter, Hilton, & Rubenstein, 1987b, p. 230)The process of changing divisions with large numbers into divisions with smaller numbers by considering them as equal ratios is explained in relation to a story context (analogous to the one in the fourth-grade book) in which problems need to be solved in this way (Willoughby et al., 1987b, p. 203) and teachers are told that "The material in this lesson is essential for the work in subsequent lessons. If any students are having difficulty it would be wise to extend this lesson to two or more days, using the extra teaching outlined below" (Willoughby et al., 1987b, p. 230). The extra teaching that is recommended focuses on students understanding of place-value relationships, and the use of play money, base-ten materials, and place-value charts are recommended.These two textbooks provide an interesting contrast in the attention that is given to the connection between the mathematical structure of long division and the conventional procedure. The justification for the steps in the procedure are modeled in the Real Math version by using trades among different denominations of money to represent the decomposition of the dividend. In the Mathematics Unlimited version, the decomposition is presented as a series of mechanical steps with no justification. Real Math
eBook - PDFSources in the Development of Mathematics
Series and Products from the Fifteenth to the Twenty-first Century
- Ranjan Roy(Author)
- 2011(Publication Date)
- Cambridge University Press(Publisher)
In these symbolic inte- gration methods, the problem of factorization is replaced by the much more accessible problems of obtaining the greatest common denominators and/or resultants of polyno- mials. These last procedures in turn require polynomial division and the elimination of variables. Contributors to symbolic integration are many, including M. Bronstein, R. Risch, and M. F. Singer. 13.2 Newton’s 1666 Basic Integrals In the beginning sections of his October 1666 tract on calculus, Newton tackled the problems of finding the areas under the curves y = 1/(c + x) and y = c/(a + bx 2 ), equivalent to evaluating integrals of those functions. Recall, however, that seventeenth- century mathematicians thought in terms of curves, even those defined by equations, rather than functions. The variables in an equation were regarded as quantities or magni- tudes on the same footing, rather than dependent and independent variables. Newton’s two integrals were the building blocks for the more general integrals of rational func- tions. It is interesting to read what he said about these integrals. He first noted the rule that if q p = ax m n , then y = na m + n x m+n n . Note that in Newton’s 1690s notation q/p was written as ˙ y/ ˙ x , whereas Leibniz wrote dy/dx . Newton next observed: Soe [so] if a x = q p . Then is a 0 x 0 = y . soe y t [that] y is infinite. But note y t in this case x & y increase in y e [the] same proportion y t numbers & their logarithmes doe [do], y being like a logarithme added to an infinite number a 0 . [That is, x 0 a t dt = a ln x − a ln 0 = a ln x + a 0 .] But if x bee diminished by c, as if a c+x = q p , y is also diminished by y e infinite number a 0 c 0 & becomes finite like a logarithme of y e number x . & so x being given, y may bee mechanically found by a Table of logarithmes, as shall be hereafter showne.
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