A LEAST SQUARES APPROXIMATE SOLUTION TO POLYNOMIAL EQUATIONS OVER THE REAL NUMBERS

A LEAST SQUARES APPROXIMATE SOLUTION TO POLYNOMIAL EQUATIONS OVER THE REALNUMBERSAuthor(s): GEORGE POOLESource: The Mathematics Teacher, Vol. 70, No. 4 (APRIL 1977), pp. 326-330Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/27960839 .

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A LEAST SQUARES APPROXIMATE SOLUTION TO POLYNOMIAL EQUATIONS OVER THE REAL NUMBERS

Solutions to 2 + bx + c = 0 give insight to those of y = 2 + bx + c.

By GEORGE POOLE Emporta State Collage

Emporia, Kansas 66801

MOST elementary presentations of the

field of complex numbers as an extension of

the real numbers fail to generate the neces

sary definitions adequately and inspire only

perfunctory remarks. In recent years there

has been considerable research in the area

of finding a "best approximate" or "least

squares" solution to the matrix equation Ax = b regardless of whether A is

nonsingular (or even square) or b lies in

the range of A (See Bouillon and Odell

[1971] and Poole [1973].) The basic idea is

to find a vector jc0 such that Ax0 is as close to the vector b as possible. That is, find x0 such that the length of the vector Ax0

? bis smallest.

Using this same idea, it is possible to

present an interesting geometric approach to the problem of finding the solutions of x2

+ bx + c = 0 when there are no real solu

tions. Such an approach provides splendid motivation for the study of complex num

bers, primarily considered as ordered pairs of real numbers.

The general idea is this: Suppose f(x) denotes the polynominal expression anxn +

an-xxn~l + ? ? ?

+ axx + a0 over the real

numbers and one is interested in solving the

polynominal equation f(x) = 0. The prob

lem is to determine numbers x0 such that

f(x0) = 0, realizing that the real number

system is the only system available to the

problem solver. The following presentation may be con

densed or expanded depending on the ob

jectives of the instructor. Furthermore, the student needs only a background of inter mediate algebra (or its equivalent) to un

derstand the presentation. Let R denote the real numbers.

As a point of motivation, consider the

problem of solving the polynominal equa tion

(1) 2 - 4x + 3 = 0.

Associated with this polynominal equation is the parabolic equation

(2) y = 2 - 4x + 3

whose graph is a parabola. The points at

which the graph intersects the x-axis (called ^-intercepts) are (1, 0) and (3, 0). Observe now that the first components of these two ordered pairs are 1 and 3, and that these real numbers are the solutions to the poly nominal equation in (1). This is not an

unexpected phenomenon, however, since the parabola must intersect the jc-axis

exactly when y = 0. That is, the x-intercepts

of the parabola given in (2) are found by replacing y by 0 in (2) and solving equation (1) for .

We have just observed that the following two problems are equivalent in the sense that if we can solve problem 1, then we can

solve problem 2, and conversely.

326 Mathematics Teacher



Problem 1. Find all real numbers that are solutions to the equation

(3) 2 + bx + c = 0

where b, c e R.

Problem 2. Find the first components of the points at which the graph of the para bola

(4) y = 2 + bx + c

intersects the x-axis, where b, c e R.

Now you might be asking about those situations where (3) does not have any real

solutions, or those where the parabola in

(4) do?s not intersect the x-axis. The dis cussion becomes more interesting at this

point. How can w? rephrase problem 2 so as to include not only the cases where the

parabolas intersect the x-axis, but also the cases where they do not intersect the x-axis?

Problem 3. Find the points on the graph of the parabola given in (4) that lie closest to the x-axis. That is, find those points (*o, yo) that satisfy equation (4) and such that l^o I is smallest (where | | denotes ab solute value).

To see that 3 is the kind of problem we are interested in, we first must agree to the

following convention: the ordered pairs of the forra ( , 0) where R will represent the set of real numbers ((x, 0) being identi fied with x). That is, the points on the x axis will represent the set of real numbers. In this way, solutions to equation (1) are

(1,0) and (3, 0) rather than 1 and 3. Now both problem 1 and problem 2 may be solved by solving problem 3.

To illustrate these ideas consider the fol

lowing two polynomial equations: 2 - 3x + 2 = 0 -x2 + 2x - 1 = 0

Their corresponding parabolic equations and graphs are pictured in figures 1 and 2.

In each example, the parabola intersects the x-axis, and these points of intersection are called the solutions to the corre

sponding polynomial equation. In the first case the solutions are (1, 0) and (2, 0).

Fig. I. y = 2 - 3x + 2

Fig. 2. y = - 2 + 2x - 1

In the second case the only solution is

(1, 0). Observe further that these solutions answer the question: Which points on the

graph of each parabola lie closest to the x axis? In both these examples, the closest

points lie on the x-axis and occur when

y = o. Of course, not all parabolas intersect the

x-axis. For example, find the solutions to 2 + 1 = 0 in R. By using the approach of

problem 3 we wish to find the points on the

graph of the parabola y - x2 + 1 that lie

closest to the x-axis (fig. 3). The closest

Fig. 3. y = 2 + 1

April 1977 327



point is (0, 1 ) and does not occur when y =

0, since the graph does not intersect the x

axis. This is equivalent to the statement

that y = 2 + 1 has no solution in R when

y - 0. However, problem 3 gives us an ap

proach to solving the equation x2 + 1 = 0.

Even though the parabola y = 2 + 1

does not intersect the x-axis (that is, x2 + 1 = 0 is not solvable in R) we may in some

sense consider the point on the graph of

y = 2 + 1 closest to the x-axis (a rep

resentation of R) as that point which is

closest to a real solution. That is, it is the

"best approximation" to a real solution of

the polynominal equation x2 + 1 = 0 we

could hope for. This "solution" (0, 1) of 2 + 1 = 0 will be identified eventually as

Prior to beginning a more general dis

cussion, observe that solving the equation ax2 + bx + c = 0 (a 0) is equivalent to

solving the equation x2 + b^x + c0 = 0

where b0 =

^, c0 = ^.

Therefore we shall

consider equations only of the form x2 + bx

+ c = 0. Before stating a definition concerning the

solutions of the polynominal equation

(5) 2 + bx + c = 0,

we consider a few properties of the corre

sponding parabolic equation

(6) y = 2 + bx + c\

whose graph is a parabola that opens up. First, if the parabola in (6) intersects the

Fig. 4. y = 2 - 3x + 2

x-axis (in one or two points), then the cor

responding hyperbola

(7) y2 = x2 + bx + c

intersects the x-axis in the same points. Therefore, the solutions (x-intercepts) of

(5), (6), and (7) are all the same. For ex

ample, consider the equations y = 2 ? 3x

+ 2 and y2 = x2 - 3x + 2, whose graphs

and intercepts are indicated in figures 4

and 5.

Second, suppose that the parabola de scribed in (6) does not intersect the x-axis. As we have suggested, a candidate for a

solution to (5) is the lowest point on the

graph of (6) that is the closest point to the x-axis. However, we need to revise our sug gestion, and a compromise is now in order.

Whatever we decide should be a solution to x2 + bx + c = 0 should also be a solution to -x2 - bx - c = 0, since both equations are simultaneously zero. The two corre

sponding parabolas are, respectively, y = 2

+ bx + c and y = ? 2 ? bx ? c. These may

be rewritten as

y = x2 -h bx -h c

?y = 2

H- bx -h c

whose graphs may take one of the three

forms pictured in figure 6.

Since we are interested in the closest

point of each parabola to the x-axis, we are

interested in minimizing the value of | j>| in

each case. The compromise is to accom

plish this simultaneously by finding the

closest points to the x-axis of the hyperbola

Fig. 5. y2 = 2 - 3x + 2




Fig. 6

(8) y2 = 2 + bx + c.

It must be noted that this compromise alters our original suggestion of "closest points." The "closest points" of (6) and the "closest

points" of (8) are identical only when x2 +

bx + c = 0 has solutions in R (that is, y =

2 + bx + c intersects the x-axis). However, the compromise is worth it, as we shall see.

In addition, observe that if (x0, yo) is a

closest point on the graph of (8) to the x

axis, then so is (x0, _.Vo). Since j>0 0, these

points are different, and we shall eventually refer to these two points as complex con

jugates. We are now ready to define what we

mean by solutions to the equation in (5).

Definition 1: The solutions of the equation in (5) are the points on the graph of

(5') / = X2 + bx + c

that lie closest to the x-axis.

The task of finding these closest points is

easy. (Readers who have had experience in

the calculus will realize the simplicity of this problem: Calculate the critical values of y2 = /(x) [where /'(x) is 0] and determine if a relative maximum or relative minimum

point occurs at these critical values. What is

the relative maximum or minimum point in

each case?) First, consider the equation (5'). By completing the square in we can

analyze the graph of y2 = x2 + bx + c:

y2 = x2 + bx + c

.(,+ta+?)+(,-s)

Since the first term on the right is

nonnegative and the second term is a con

stant, y2 is smallest when the first term is

zero. That is, y2 is smallest when = ?y.

In this case / = (4c ~

?2).

Now recall that if (5) has real solutions

?that is, the graph of (6) intersects the jc

axis?then the graph of (8) intersects thex

axis and the solutions occur when y = 0.

On the other hand, if (5) has no real

solutions, according to definition 1 we are

going to interpret the closest points on the

graph of (5') to the x-axis as solutions to

(5). Therefore, in view of our observations

above, the closest points (solutions) occur

, b c. (4c -

b2) . when =-? Since v--A-1 is posi a 4 r

twe (parabola opens up and does not inter

sect x-axis),

J4c- b2 J4c - b2 ^

= ^__or>;=_>L_

April 1977 329



That is, the solutions of (5) when no rea solutions exist are

(_ |, and

(- |,_^ ) To illustrate these concepts, consider the

following three equations and correspond ing hyperbolic graphs:

(a) . 2 - 2.v = 0 - y2 = x2 - 2x

Fig. 7

Solutions: (0, 0), (2, 0)

(b) (x -

3)2 = 0 - ra = (.

- 3)2

Fig. 8

Solution: (3, 0)

(c) 2 + + 1 = O -> y2

= 2 + : + 1

Fig. 9

? , (-if ).(}-#)

The foregoing discussion can be general ized to include polynominal equations over

R of higher degree, but the abbreviated dis cussion was presented to provide motiva tion for considering another number sys tem. This number system consists of ordered pairs of real numbers and shall be called the complex numbers. The ordered

pairs of real numbers ( 0, y0) represent so lutions of polynomial equations in the sense described in definition 1. x0 is called the real part of the complex number (x0, y0) and y0 is called the imaginary part. Also, in the complex plane (0, 1 ) is often denoted by "/," and (x0, >>o), C*o, _> >) are called com

plex conjugates.

references

Bouillon, T. L., and P. L. Odell. Generalized Inverse Matrices. New York: Wiley, 1971.

Poole, George. "Geometry of Minimum Norm Least

Squares Solutions of Matrix Equations." Delta (Fall 1973): 12-17.

-. An Algebraic Development of the Real Num bers. Emporia, Kansas: Emporia Kansas State Col

lege Printing Office, 1974.




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A LEAST SQUARES APPROXIMATE SOLUTION TO POLYNOMIAL EQUATIONS OVER THE REAL NUMBERS