The coefficient of z in the new equation is the remainder from dividing the quotient just obtained by x — a. The coefficients of the higher powers of z are the remainders from dividing the successive quotients obtained by x-a. EXAMPLE. Form the equation whose roots are 2 less than the roots of x4 — 2 x3- 4 x2 + x − 1 = 0. The divisions required by the rule we carry out synthetically (p. 169). 181. Graphical interpretation of decreasing roots. If an equation has roots a units less than those of another equation, if a is positive its intersections with the X axis or with any line parallel to the X axis are a units to the left of the corresponding intersections of the first equation. It is, in fact, the same curve, excepting that the Y axis is moved a units to the right. If a is negative, the Y axis is moved to the left. EXERCISES Plot, decrease the roots by a units, and plot the new axes. YA .3. 182. Location principle. If when plotting an equation y = f(x) the value x = a gives the corresponding value of y positive and equal to c, while the value x = b gives the corresponding value of y negative, say equal to d, then the point on the curve x = a, y = c is above the X axis, and the point on the curve x= b, y d is below the X axis. If our curve is unbroken, it must then cross the X axis at least once between the values xa and x = b, and hence the equation must have a root between those values of x. The shorter we can determine this == 0 interval a to b the more accurately we can find the root of the equation. This property of unbrokenness or continuity of the graph of y = a。x2 + а1x2-1 + ·+a, we assume. We assume then the following LOCATION PRINCIPLE. When for two real unequal values of x, say x = a and x=b, the value of y = f(x) has opposite signs, the equation f(x)=0 has a real root between a and b. ILLUSTRATION. The equation f(x) = x2 + 3 x − 5 = 0 has a root between 1 and 2. Since ƒ(1)=-1, f(2) = 9. 183. Approximate calculation of roots by Horner's method. We are now in a position to compute to any required degree of accuracy the real roots of an equation. Consider for example the equation x2 + 3x · 20 = 0. (1) the origin is at the less of the two integral values between which we know the root lies. Here we decrease the roots of (1) by 2, The equation whose roots are decreased by 2 is x3 +6 x2 + 15 x − 6 = 0. (2) We know that (2) has a root between 0 and 1, since equation (1) has a root between 2 and 3. From the graph we can estimate the position of the root. Having made an estimate, say .3, it is necessary to verify the estimate and determine by synthetic division precisely between which tenths the root lies. Thus, trying .3, we obtain 1 + 6.0 + 15.00 – 6.000 .3 +0.3 +1.89 + 5.067 1 + 6.3 + 16.89 - 0.933 which shows that for x = .3 the curve is below the X axis, hence the root is greater than .3. But we are not justified in assuming that the root is between 3 and 4 until we have substituted .4 for x. This we proceed to do. Since the value of y is positive for x = .4, the location principle shows that (2) has a root between .3 and .4, that is, (1) has a root between 2.3 and 2.4. To find the root correct to two decimal places, move the origin up to the lesser of the two numbers between which the root is now known to lie. The new equation will have a root between 0 and .1. This process is performed as follows: This equation has a root between 0 and .1. We can find an approximate value of the hundredths place of the root by solving = the linear equation 18.87 x .933 0, obtained from (3) by dropping all but the term in x and the constant term. This suggestion must be verified by synthetic division to determine between what hundredths a root of (3) actually lies. Thus the curve is below the X axis at x= .04 and hence the root is greater than .04.. We must not assume that the root is between .04 and .05 without determining that the curve is above the X axis at x = .05. Thus the curve is above the X axis at x = .05. By the location principle (3) has a root between .04 and .05, that is, (1) has a root between 2.34 and 2.35. We say that the root 2.34 is correct to two decimal places. If a greater degree of precision is desired, the process may be continued and the root found correct to any required number of decimal places. The foregoing process affords the following RULE. Plot the equation. Apply the location principle to determine between what consecutive positive integral values a root lies. Decrease the roots of the equation by the lesser of the two integral values between which the root lies. Estimate from the plot the nearest tenth to which the root of the new equation lies, and determine by synthetic division precisely the successive tenths between which the root lies. Decrease the roots of this equation by the lesser of the two tenths between which the root lies, and estimate the root to the nearest hundredth by solving the last two terms as a linear equation. |