tailieunhanh - Handbook of mathematics for engineers and scienteists part 29

An equilateral (or regular) triangle is a triangle with all three sides equal (Fig. 3. All angles of an equilateral triangle are equal to 60c. In an equilateral triangle, the circum-radius R and the inradius r satisfy the relation R = 2r. | 164 Algebra 2o. A constant K is called an upper bound for the real roots of equation or the polynomial Pn x if equation has no real roots greater than or equal to K in a similar way one defines a lower and an upper bound for positive and negative roots of an equation or the corresponding polynomial. Let K1 be an upper bound for the positive roots of the polynomial Pn x K2 be an upper bound for the positive roots of the polynomial Pn -x K3 0 be an upper bound for the positive roots of the polynomial XnPn 1 x K4 0 be an upper bound for the positive roots of the polynomial xnPn -1 x . Then all nonzero real roots of the polynomial Pn x if they exist belong to the intervals -K2 -1 K and 1 K3 Ki . Next we describe three methods for finding upper bounds for positive roots of a polynomial. Maclaurin method. Suppose that the first m leading coefficients of the polynomial are nonnegative . an 0 an-1 0 . an-m 1 0 and the next coefficient is negative an-m 0. Then K 1 1 m an is an upper bound for the positive roots of this polynomial where B is the largest of the absolute values of negative coefficients of Pn x . Example 3. Consider the fourth-degree equation from Example 2. In this case m 2 B 36 and formula yields K K1 1 36 9 1 2 3. Now consider the polynomial P4 -x 9x4 -9x2 36x 1. Its positive roots has the upper bound K2 1 9 9 1 2 2. For the polynomial x4P4 1 x x4 -36x3 -9x2 9 we have m 1 K3 1 36 37. Finally for the polynomial x4P4 -1 x x4 36x3 - 9x2 9 we have m 2 k4 1 91 2 4. Thus if P4 x has real roots they must belong to the intervals -2 -1 4 and 1 37 3 . Newton method. Suppose that for x c the polynomial Pn x and all its derivatives Pn x . Pnn x take positive values. Then c is an upper bound for the positive roots of Pn x . Example 4. Consider the polynomial from Example 2 and calculate the derivatives P4 x 9x4 - 9x2 - 36x 1 P4 x 36x3 - 18x - 36 P4 x 108x2 - 18 P4 x 216x P4 x 216. It is easy to check that for x 2 this polynomial