tailieunhanh - Handbook of mathematics for engineers and scienteists part 28

In a right triangle, the length of the median mc drawn from the vertex of the right angle coincides with the circumradius B and is equal to half the length of the hypotenuse c7 mc = R = jc. The inradius is given by the formula r = -(a -\-b-c). The area of the right triangle is S = aha = -a5(see also Paragraphs to ). | . Polynomials and Algebraic Equations 157 To find the coefficients b0 . bn of this expansion one first divides f x by x - c with remainder. The remainder is bo and the quotient is some polynomial g0 x . Then one divides g0 x by x - c with remainder. The remainder is b1 and the quotient is some polynomial gi x . Then one divides g1 x by x - c obtaining the coefficient b2 as the remainder etc. It is convenient to perform the computations by Horner s scheme see Paragraph . Example 4. Expand the polynomial f x x4 - 5x3 - 3x2 9 in powers of the difference x - 3 c 3 . We write out Horner s scheme where the first row contains the coefficients of the polynomial f x the second row contains the coefficients of the quotient g0 x and the remainder b0 obtained when dividing f x by x - 3 the third row contains the coefficients of the quotient g1 x and the remainder b1 obtained when dividing g0 x by x - 3 etc. 11-5-30 9 3 1 -2 -9 -27 -72 1 1 -6 -45 1 4 6 1 7 1 Thus the expansion of f x in powers of x - 3 has the form f x x - 3 4 7 x - 3 3 6 x - 3 2 - 45 x - 3 - 72. The coefficients in the expansion of a polynomial f x in powers of the difference x - c are related to the values of the polynomial and its derivatives at x c by the formulas b _ f .y b _ f c b _ f x c b _ f n c b0 f c b1 . . b2 o. . bn . 1 2 n where the derivative of a polynomial f x an xn an-1xn-1 a1x x0 with real or complex coefficients a0 . an is the polynomial f x x nanxn-1 n-1 an-1xn-2 a1 f c x f x X etc. Thus Horner s scheme permits one to find the values of the derivatives of the polynomial f x at x c. Example 5. In Example 4 the values of the derivatives of the polynomial f x at x 3 are f 3 -72 f 3 -45X1 -45 f 3 6x2 12 f 3 7x3 42 fIV 3 1x4 24. The expansion of a polynomial in powers of x - c can be used to compute the partial fraction decomposition of a rational function whose denominator is a power of a linear binomial. Example 6. Find the partial fraction decomposition of the rational function x .