# How to Solve division algorithm

###### how to prove division algorithm, how to verify division algorithm,how division algorithm works, how to do a division algorithm, how to write a division algorithm pdf free download
Prof.EvanBaros,United Kingdom,Teacher
Published Date:26-07-2017
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Division algorithm MATH 1510 Lili Shen Dividing Deﬁnition Polynomials Rational If P(x) and D(x) are polynomials, with D(x) =6 0, then there Functions exist unique polynomials Q(x) and R(x), where R(x) is either 0 or of degree less than the degree of D(x), such that P(x) = D(x)Q(x)+R(x): P(x) and Q(x) are called the dividend and divisor, respectively. Q(x) is the quotient, and R(x) is the remainder.Long division of polynomials MATH 1510 Lili Shen Dividing Polynomials Rational Functions Example 2 Divide 6x 26x + 12 by x 4.Long division of polynomials MATH 1510 Lili Shen Solution. Dividing Polynomials 6x 2 Rational  Functions 2 x 4 6x 26x + 12 2 6x + 24x 2x + 12 2x 8 4 Thus 2 6x 26x + 12 = (x 4)(6x 2)+ 4:Synthetic division MATH 1510 Lili Shen Dividing Polynomials Rational Functions Example 3 2 Divide 2x 7x + 5 by x 3.Synthetic division MATH 1510 Lili Shen Dividing Solution. Polynomials Rational 2 7 0 5 Functions 3 6 3 9 2 1 3 4 Thus 3 2 2 2x 7x + 5 = (x 3)(2x x 3) 4:The remainder theorem MATH 1510 Lili Shen Dividing Polynomials Rational Functions Theorem If the polynomial P(x) is divided by xc, then the remainder is the value P(c).The remainder theorem MATH 1510 Lili Shen Dividing Polynomials Rational Functions Example 5 4 3 Let P(x) = 3x + 5x 4x + 7x + 3. (1) Find the quotient and remainder when P(x) is divided by x + 2. (2) Use the remainder theorem to ﬁnd P(2).The remainder theorem MATH 1510 Lili Shen Solution. Dividing Polynomials (1) Rational Functions 3 5 4 0 7 3 2 6 2 4 8 2 3 1 2 4 1 5 4 3 2 Thus the quotient is 3x x 2x + 4x 1, and the remainder is 5. (2) P(2) = 5.The factor theorem MATH 1510 Lili Shen Dividing Polynomials Rational Functions Theorem P(c) = 0 if and only if xc is a factor of P(x).The factor theorem MATH 1510 Lili Shen Dividing Polynomials Rational Functions Example 3 Factorize P(x) = x 7x + 6.The factor theorem MATH 1510 Lili Shen Solution. Dividing Polynomials Since P(1) = 0, we know that x 1 is a factor of P(x). Rational Functions 1 0 7 6 1 1 1 6 1 1 6 0 Thus 2 P(x) = (x 1)(x +x 6) = (x 1)(x 2)(x + 3):Outline MATH 1510 Lili Shen Dividing Polynomials Rational Functions 1 Dividing Polynomials 2 Rational FunctionsRational functions MATH 1510 Lili Shen Dividing A rational function is a function of the form Polynomials Rational P(x) Functions r(x) = ; Q(x) where P and Q are polynomials. Usually we assume that P(x) and Q(x) have no common divisors. The domain of a rational function consists of all real numbers x except those for which the denominator Q(x) = 0.A simple rational function MATH 1510 Lili Shen Dividing Example Polynomials Rational 1 Functions Consider the rational function f(x) = . The domain and x the range of f are bothfxj x6= 0g. The behavior of f can be described as: f(x)1 as x 0 ; + f(x)1 as x 0 ; f(x) 0 as x1 or x1.A simple rational function MATH 1510 Lili Shen The graph of f is sketched as: Dividing Polynomials y Rational 4 Functions 2 x 4 2 2 4 2 4Vertical and horizontal asymptotes MATH 1510 Lili Shen Dividing Polynomials Rational The lines x = 0 and y = 0 are respectively the vertical Functions 1 asymptote and the horizontal asymptote of f(x) = . x Informally speaking, an asymptote of a function is a line to which the graph of the function gets closer and closer as one travels along that line.Vertical and horizontal asymptotes MATH 1510 Lili Shen Dividing Polynomials Rational Functions Deﬁnition The line x = a is a vertical asymptote of the function + y = f(x) if y1 or y1 as x a or x a . The line y = b is a horizontal asymptote of the function y = f(x) if y b as x1 or x1.Transformations of y= 1=x MATH 1510 Lili Shen Dividing Polynomials A rational function of the form Rational Functions ax +b r(x) = cx +d can be obtained by by shifting, stretching, and/or reﬂecting the function 1 f(x) = : xTransformations of y= 1=x MATH 1510 Lili Shen Dividing Polynomials Example Rational Functions State their domain, range and asymptotes of the following functions, and sketch their graphs: 2 (1) r(x) = . x 3 3x + 5 (2) s(x) = . x + 2Transformations of y= 1=x MATH 1510 Lili Shen Dividing Polynomials Solution. Rational 1 Functions (1) Let f(x) = . Then x   2 1 r(x) = = 2 = 2f(x 3): x 3 x 3 The domain isfxj x6= 3g, and the range is fyj y =6 0g. r has vertical asymptote x = 3 and horizontal asymptote y = 0.

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