1550078481-Ordinary_Differential_Equations__Roberts_

198 Ordinary Differential Equations

learn is how to interpret results from a computer. When a p olynomial has
repeated real roots-such as, -4, -4- small , erroneous imaginary p arts are
often computed. Since 1.15 x 10-^11 is small compared to 4, we can easily
surmise t hat the actual zeros may b e - 4 and -4. The fourth zero is 2 - 3i
and the fifth is -1 + 6i. We conclude t hat the sixth zero of the polynomial
is probably 5 instead of the computed value of 5 + 2.419410 x 10 -^16 i. The
seventh zero is -1 - 6i.

THE ZEROS OF THIS POLYNOMIAL ARE

ZERO REAL. PART IMAGINARY PART

l 2 .000000E+OO 3.000000E+OO

2 -4.oooooox+oo -L 153824R-ll

3 -4.000000R+OO l.153809B-ll

4 2.000000R+OO -3.000000R+OO

s -L OOOOOOR+OO 6. 000000 E+OO

6 S.OOOOOOE+OO 2.419410B-16

7 -1.000000R+OO -6.00000 0 R+OO

Figure 4 .2 Zeros of the Polynomial Equation (3)

EXAMPLE 2 Calculation of the Roots of a Polynomial

Find the roots of the polynomial equation

(4) x^3 + (-7 - 3i)x^2 + (10 + 15 i)x + 8 - 12 i = 0.

SOLUTION

We entered 3 for the degree of the polynomial equation to be solved into
POLYRTS. Then we entered the values for the coefficients. The roots of this
polynomial as calculated by POLYRTS a re displayed in Figure 4.3. The values
shown should be interpreted as i, 3 + 2i , and 4.

ZERO

l

z

3

THE ZEROS OF THIS POLYNOMIAL ARR

REAL PART

-7.41463SR-17

3.000000E+OO

4.000000E+OO

IMAGINARY PART

l.OOOOOOE+OO

Z.OOOOOOE+OO

- 4.440892E- 16

Figure 4 .3 Zeros of t he P olynomial Equation (4)