finding the complex roots of polynomials

Estimating Complex Roots of Polynomials Using Data Analysis

Finding the Complex Zeros of Polynomials In any discussion of the roots of polynomial equations at the algebra or precalculus level, one typically stresses the fact that both real and complex roots can arise and that complex roots only occur in pairs. Moreover, the real roots correspond to the zeros of the polynomial, so that they can always be found graphically or numerically to any desired degree of accuracy. Unfortunately, there is no simple way to visualize the complex roots,Invariably, several students will ask: How do you find the complex roots? My standard answer has always been a very truthful: That is a much harder issue that is well beyond the level of this course for polynomials of degree three and higher. However, if you ever need to actually find the complex roots, routines are built into many calculators (such as the TI-86) or into many software packages. But, for purposes of this course, that is something that you dont have to worry about.

Nevertheless, this response has always been unsatisfying to me. Perhaps the most disturbing reason is that I believe it is essential to encourage creative thinking on the part of my students and this answer, although literally true, tends to turn off that creative questioning by telling the students who have come up with the spark of interest that the methods are way beyond them. While that may be true in the sense of mathematical sophistication, it likely suggests to the students that they arent smart enough to appreciate the mathematical methods or ideas involved.

By far, the simplest and most direct way of finding complex roots is via the quadratic formula for quadratic equations. It was essentially known to the ancient Babylonians, some 3000 years ago. In the 1530s, the Italian mathematician Niccol Tartaglia discovered a formula (it takes about half of a printed page to display) for finding the three roots of any cubic equation, though he attempted to keep the result a secret. Several years later, the Italian mathematician Lodovico Ferrari discovered a method for solving for the four roots of any quartic equation (this takes two full printed pages). In 1824, the Norwegian mathematician Niels Abel proved that it is not possible for any such formula to exist for all polynomials of degree five or higher; it became known as the Abel Impossibility Theorem. More recently, numerical analysts have developed numerical techniques for approximating the complex roots of any polynomial equation to any desired degree of accuracy.In the present article, we introduce ways of estimating the complex zeros of many polynomials that are natural outgrowths of ideas and techniques that are a part of algebra and precalculus courses. There are some limitations to this approach, which we will also discuss. But, they provide an effective way to respond to the students question and simultaneously provide a nice way to tie together a variety of other ideas using a combination of graphical, numerical, algebraic, and technological approaches to reinforce key mathematical ideas.Complex Zeros of Cubics Consider the cubic polynomial which has three zeros, and at least one must be real. As such, one can always find that real zero graphically or numerically without worrying about whether the polynomial is easily factorable. To facilitate this investigation, the author has developed an Excel spreadsheet that can be downloaded from for use by teachers for classroom demonstrations or by students for individual or small group investigations. The spreadsheet allows the user to vary the four parameters a, b, c, and d using sliders, so that the results, both graphical and numerical, appear instantaneously. For instance, suppose you select , so that A fifth slider allows the user to trace along the curve of the polynomial and the associated point x = x0 is used as an approximation to a real zero. In particular, it turns out that is fairly close to the single real zero of this cubic; note that When C(x) is divided by , the result is with a remainder of 1. In general, if C(x) is divided by the result is with a remainder of

The spreadsheet ignores the remainder and so approximates

See Figure 1, where the graph of C(x) is in blue and that of the approximation is in red; they are clearly very close to one another across the entire interval [-4, 4].

If , in this case, is sufficiently close to a real zero of C(x), then will be close to a quadratic factor of C(x) and its zeros (whether real or complex) will be close to the other two zeros of C(x). In this case, the other zeros are rounded to two decimal places. Actually, from the quadratic formula, we get exactly. We can improve on this slightly by looking, graphically, for a value for x0 that is still closer to the real zero; in particular, with x0 = 0.56, we have and the corresponding factorization is approximately

The associated zeros are then We note that the graphs of the two functions are essentially indistinguishable, so that the approximation is effectively a perfect match. On the other hand, the farther that x0 is from a real zero, the further apart the two graphs become, as illustrated in Figure 2 based on x0 = 4, where the approximating factored polynomial is and the associated complex zeros are Clearly, these cannot be too accurate.Complex Zeros of Quartics We now extend the approach from the previous section to quartic polynomials of the form For convenience, we consider only quartics whose leading coefficient is 1. The possible cases are all four zeros are real, two are real and two are complex, and two pairs of complex zeros. Clearly, any real zeros can be found graphically and, if there is only a single pair of complex zeros, they can theoretically be found by dividing Q(x) by each of the two corresponding linear factors and applying the quadratic formula. Thus, the case we focus on is where there are only complex zeros.

As with the cubic investigation, we also have an Excel spreadsheet to handle the work and allow us to concentrate on dynamic explorations. It is also available for download from . To use it, you have to enter the values of the four coefficients a, b, c, and d using sliders. In addition, the basic idea is to try to estimate a quadratic factor of the form x2 + x + , where the two parameters are also selected using sliders to provide seemingly instantaneous responses. The spreadsheet then displays two graphs one is the graph of the original quartic with a tracing point highlighted and the second shows both the original quartic and the product of the estimated quadratic factor y = x2 + x + and the quotient of Q(x) divided by this quadratic, again with the remainder being ignored. As one varies the parameters and the product of the approximate quadratic factors moves closer to or further away from the original quartic curve. The idea is to find a combination of and that produces as close as match as possible. If the two curves are indistinguishable, then one is very close to the correct factorization and all four of the complex roots (if that is the case) can be found from the quadratic formula.

To illustrate such an investigation, suppose we take Q(x) = x4 + 0x3 + 5x2 + 0x + 4, which happens to factor as Q(x) = (x2 + 1)( x2 + 4), so that the four zeros are trivial to find. Figure 3 shows the two graphs with = -2 and ( = -1; this is obviously a very poor choice since not only are the two curves not particularly close, but also their behaviors are quite different with the supposed factorization function having two real roots. In particular, the factorization is

and the associated zeros are (the latter two being rounded to two decimal places). On the other hand, Figure 4 shows the results if we select = 0 and ( = 2; the two graphs appear almost indistinguishable. The corresponding factorization is

and the associated approximate zeros are Clearly, these approximate zeros are not particularly close to the correct values of i and 2i.

The fact that the two graphs in Figure 4 appear almost indistinguishable is obviously misleading; it is due to the relatively large vertical scale. As such, we need a way to measure how close the two functions are to one another. We use the Sum of the Squares of the vertical distances, which is the most commonly used measure of how well a function fits a set of data. In particular, the spreadsheet uses 200 uniformly spaced points across the interval to calculate the sum of the squares. For the current case with = 0 and ( = 2, the value is 804.00; in the previous case with = -2 and ( = -1, the value is 563432.48. If we vary the parameter slightly about = 0, we find that there is no noticeable decrease in the Sum of the Squares it only increases from the 804 in either direction. However, if we vary (, we find that the Sum of the Squares decreases as ( decreases and actually becomes 0 when ( = 1. Thus, the choice of = 0 and ( = 1 produces a perfect fit across the interval and the quadratic factorization is

as expected.

The identical analysis can be applied to any quartic polynomial, including those that have real zeros. However, one should not expect that the Sum of the Squares will always reduce to a minimum of 0. If the values chosen are not within the limits of accuracy (two-decimal place) of the spreadsheet program, we will not get a perfect fit and would have to settle for the best possible fit instead. Presumably, though, the values for the zeros will be fairly close to the true zeros.Clearly, additional spreadsheets of this type could be developed to allow explorations of the zeros of polynomials of higher degree. However, they would more likely become computational tools rather than investigatory activities that reinforce important mathematical concepts and methods in students minds Bairstows Method for Complex Zeros In practice, the problem of solving for the complex zeros of a polynomial is accomplished using numerical methods. Perhaps the most widely used is Bairstows Method, which was developed by the aeronautical engineer Leonard Bairstow for his 1920 book Applied Aerodynamics and so indicates that complex numbers arise far more widely in applications than most students tend to believe. The method is based on the Newton-Raphson Method for solving for the roots of f(x, y) = 0; this is the two-variable extension of the usual Newtons Method from calculus. Bairstows Method starts with an initial estimate (often a guess) of a possible quadratic factor x2 + x + for a polynomial of degree n. It then successively adjusts the coefficients and ( until its zeros are also the zeros of the polynomial P(x). The process is one in which the convergence is eventually extremely rapid; it is described as quadratic convergence and has the property that once the first decimal places have been determined, each successive iteration effectively doubles the number of correct decimal places. While we will not go into the details here, the interested reader is directed to [1], say, for more information on Bairstows Method.

The author has developed another Excel spreadsheet that is a graphical and numerical implementation of Bairstows Method. It also is available from . You enter the coefficients of a polynomial of degree between 3 and 8 and the initial guesses for the coefficients and ( of a potential quadratic factor . The program then performs the first five iterations of the method, draws the graphs of the successive approximations to the quadratic factor, and displays the equations of each of the approximations. When and ( are chosen reasonably close to the correct values, the convergence is extremely fast; the sequence of curves drawn very rapidly become indistinguishable from one another and the coefficients of the successive quadratics similarly seem to converge to the number of decimals places displayed.

For instance, consider From the graph in Figure 5, it seems that this polynomial has no real zeros and hence two pairs of complex conjugate zeros; a little exploration with larger windows provides further verification of this. Since this polynomial factors easily, we know that the zeros are and If we apply Bairstows Method starting near the quadratic with = ( = -1, we get the following sequence of successive quadratic approximations:

All results are shown to four decimal places. The associated graphs of these six quadratic functions, drawn with matching colors, are shown in Figure 6. The successive graphs tend to suggest that the successive curves may be converging (the last three are relatively close together), though the coefficients shown in the above formulas may not make that eminently clear. To continue the process, we continue the procedure with the spreadsheet, using = -0.0642 and ( = -0.6804 from the n = 4 case so that we have the results for n = 5 for comparison. The associated results are

Obviously, the process has converged, and quite rapidly, to give two complex zeros . In fact, correct to 10 decimal places, the last two quadratics are actually and and it becomes clear that the number of correct decimal places has roughly doubled in each of the two coefficients. The associated graphs of the successive quadratic approximations in Figure 7 show that the curves are virtually indistinguishable.If the sequence of quadratics does not seem to converge, it is a very simple matter to vary the starting coefficients and see the effects essentially instantaneously. The technique thus becomes an effective exploratory one as students experiment with different choices of the starting values to see how good they are.

Some authors suggest that, if the leading coefficient of the original polynomial cn is 1, then a good choice for and ( are cn-1 and cn-2, respectively. If cn 1, then a good choice for and ( are cn-1/ cn and cn-2/cn, respectively. Incidentally, if the initial choices are poorly chosen, then the convergence can be extremely slow as the successive values seem to meander all over the map, at least until the first decimal places are determined and then the convergence becomes quadratic. Moreover, Bairstows Method works for any polynomial, provided that there are no repeated quadratic factors, such as However, the higher the degree of the polynomial, the more iteration that might be needed to get close to good starting values for and (.Student Projects I have found that individualized student projects, with the support of appropriate technology, are extremely effective in reinforcing important concepts and methods at all levels of mathematics. One approach that has been especially useful is the use of projects that are based on students social security numbers; typically these involve using, say 123-45-6789, to form the coefficients of an personal polynomial where students might be required to find all the real zeros (at the algebra or precalculus level) or perform a complete max-min analysis (in calculus). For either project, it is usually desirable to have the students alternate the signs of the coefficients, say with to get more action because such polynomials tend to have a preponderance of complex zeros (much to the surprise of the students). In the present context, where the objective is to approximate the complex zeros, it likely would not be necessary to modify the coefficients. As such, the project would require the students to (1) determine the number of real and complex zeros (usually from an analysis of the graph), (2) determine all the real zeros to, say, six decimal places (using a combination of graphical and computational tools), and (3) determine all the complex zeros (using the Excel implementation of Bairstows Method). Because the polynomials are based on something individual for each student, they tend to become very possessive of my polynomial and accordingly much more interested in interested in determining its behavior. And, of course, that personal interest translates into a high level of motivation, especially since much of the work entails the use of software (usually Excel spreadsheets that I provide) that allows the students to conduct exploratory investigations, see the results immediately, and have the ability to print out the resulting graphs and tables quickly and easily. In turn, this provides them with the documentation that forms the data for an individualized project report. Not only do such projects count reasonably heavily toward grades, they occasionally form the basis for presentations at math fairs and similar competitions.

However, as we become more conscious of privacy issues, it might be desirable to base individualized polynomials on students telephone numbers or other personalized information rather than on social security numbers. Reference

Hamming, R. W., Numerical Methods for Scientists and Engineers, 2nd Ed., Dover, 1987, ISBN-10: 0486652416

Figure 1: The Cubic vs. the Approximate Factorization with x0 = 1

Figure 2: The Cubic vs. the Approximate Factorization with x0 = 4

Figure 3: The Quartic vs. the Approximate Factorization with = -2, ( = -1

Figure 4: The Quartic vs. the Approximate Factorization with = 0, ( = 2

Figure 5: The graph of y = x4 + 5x2 + 4 on [-3, 3]

Figure 6: Bairstows Method starting near x2 + x + 1

Figure 7: Bairstows Method starting near x2+0.0642x+0.6804_1472024617.unknown

_1472455927.unknown

_1472457037.unknown

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_1472536824.xlsChart3

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Sheet1

Bairstow's Method

for Finding the Complex Roots

of a Polynomial

This DIGMath illustrates and applies

Bairstow's Method for finding the complex roots

of any desired polynomial (up to 8th degree)

y = c8 x8+c7 x7+c6 x6+c5 x5+c4 x4+c3 x3+c2 x2+c1 x+c0

where c0, c1, c2, c3, c4, c5, c6, c7, c8 are any coefficients

on any interval from x = xMin to x = xMax.

First, select the degree ( k = 3 to 8) of your polynomial:

44

Enter an initial guess for a quadratic factor: y = x2 - A x - B

Next, enter the value for each of the coefficients in

y = c0 + c1x + c2x^2 + c3x^3+ c4x^4First,A =-1

(Use 0's if you don't want some of the terms.)

ThenB =-1

c0 =4

c1 =0

c2 =5The successive Bairstow quadratic approximations are:

c3 =0n = 0:Q(x) = x^2 + 1x + 1

c4 =14

015n = 1:Q(x) = x^2 + 0.1000x + 0.3000

026

007n = 2:Q(x) = x^2 + 0.4252x - 2.2723

028

n = 3:Q(x) = x^2 + 0.1840x - 0.2700

The graph of y = 4.00 + 0.00x + 5.00x^2 + 0.00x^3 + 1.00x^4

n = 4:Q(x) = x^2 + 0.0642x + 0.6804

Now enter the interval for x:

xMin =-2.7n = 5:Q(x) = x^2 + 0.0103x + 0.9687

xMax =2.7

The two complex zeros of the last quadratic are roughly:

x =-0.0321052054+0.9841898805i

Click each item below for suggestions and investigationsx =-0.0321052054-0.9841898805i

Item 1

Item 2

Item 3

Created by: Sheldon P. Gordon

Farmingdale StateCollege

Development of this module was supported by the

NSF's Division of Undergraduate Education

under grants DUE-0310123 and DUE-0442160.

Sheet1

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n = 0 n = 1 n = 2 n = 3 n = 4 n = 5

Sheet2

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-2.18750.7916374553-0.18402330760.2700392414

-2.1649.095823364-0.0642104108-0.6803576185deg n = 5Bairstow 3bcDenom'sNext uNext vbcDenom'sNext uNext vbcDenom'sNext uNext vbcDenom'sNext uNext vbcDenom'sNext uNext v

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-1.8934.62039841


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-0.7567.1843333993QuadraticsnP(x)

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-0.7026.7068757824-2.75.594y = c0 + c1x + c2x^2 + c3x^3+ c4x^4

-0.6756.4857191406015y = c0 + c1x + c2x^2 + c3x^3+ c4x^4+ c5x^5

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2.72796.4846085678

(1) Bairstow's Method is a numerical technique for estimating pairs of complex roots of a polynomial by finding a quadratic factor of the polynomial. To use this DIGMath graphing utility, you have to enter the coefficients of your desired polynomial in the cells above. You then have to enter the values for the interval of x values from xMin to xMax. The spreadsheet then produces the graph of the polynomial on this interval in the chart above. In addition, you enter the initial 'guesses' for the coefficients A and B in the trial quadratic factor x^2 - Ax - B, and the spreadsheet then calculates, prints, and graphs the first five approximations to a true quadratic factor.

(2) Observe how the successive approximations to the quadratic factor typically get closer and closer to one another in the graph at the top right. Simultaneously, observe how the coefficients of these successive quadratic approximations tend to get closer to one another as the process continues. Realize that, once you have the quadratic factor to the desired degree of accuracy, you can find its two roots using the quadratic formula. If the quadratic that is produced has complex roots, that means that you have estimates for the complex roots of the original polynomial. Otherwise, you have estimates for two of the real roots. Also, you can reduce the degree of the polynomial by 2 by dividing through algebraically by the quadratic factor.

(3) You should investigate what happens when you change the coefficient A and B in the initial quadratic polynomial. Do you always converge to the same quadratic factor? Does the rate of convergence depend on the coefficients you select?

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Sheet3

MBD016E6DCD.unknown

MBD000058D6.unknown

_1472539148.xlsChart2

-13434-168

-32-66

-2-36

4-30

340

The Cubic vs. the Approximate Factorization

Sheet1

Finding the Zeros

of Cubic Functions

This DIGMath module estimates the

three zeros of any cubic polynomial

y = ax3 + bx2 + cx + d

whether they are all real zeros or

one real and two complex zeros.

What is the value for a (-5 to 5):

a =160

What is the value for b (-5 to 5):

b =-320

Trace along the curve to estimate a real zero

What is the value for c (-5 to 5):x0 =4800

c =5100

then y0 =34.00

What is the value for d (-5 to 5):

d =-230The factorization of the cubic is roughly

y = (x - 4.00) (1.00x^2 + 1.00x + 9.00)

The zeros of the quadratic factor are roughly

Click each item below for suggestions and investigationsx = -0.50 + 2.96i and x = -0.50 - 2.96i

Item 1

Item 2

Item 3






1. Every cubic polynomial y = ax3 + bx2 + cx + d has precisely three zeros. Since complex zeros occur only as complex conjugate pairs, a cubic can have either three real zeros or one real zero and a pair of complex zeros. This DIGMath spreadsheet lets you find all three zeros of any cubic through a combination of graphical investigation and algebraic computation that takes place behind the scenes. Start by defining your cubic polynomial using the sliders above to select values for the four coefficients. Then use the slider to the right to trace along the cubic curve. Each point x0 is used as an approximation to a real zero of the function and the quotient of the function divided by the linear factor (x - x0) is displayed, ignoring the remainder, as an approximation to the factored form of the function.

2. The closer that x0 is to an actual real zero of the cubic, the more accurate the factorization is. In turn, the closer you are to a real zero, the closer that the zeros of the quadratic factor are to the actual zeros, either real or complex, of the original cubic polynomial.

3. We suggest that you start with a cubic that you know factors, most likely by starting with three linear factors and multiplying them out or with one linear factor and one quadratic factor and multiplying them out. This will give you the chance to see how the spreadsheet operates. Then feel free to vary the values of the parameters as much as you want and watch how the two curves in the lower chart approach each other while the quadratic term approaches the actual quadratic factor and the zeros of the approximate quadratic factor approach those of the actual quadratic factor.

Sheet1

-134-8.432

-32

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4

34

Sheet2

-13434-168

-32-66

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4-30

340


Sheet3

plotting points:xyapprox00

-4-134-16800

-2-32-66

Disc =-350-2-36

24-30

D > 04340

root1 =0.000

root2 =0.000434

Quadratic factor:

D = 0a =1

root3 =0.000b =1

root4 =0.000c =9

D < 0

root5 =-0.52.9580398915i

root6 =-0.5-2.9580398915i

MBD00000050.unknown

MBD000000A0.unknown

MBD0170A42D.unknown

MBD016FEEB8.unknown

MBD00000078.unknown

MBD00000028.unknown

_1472540100.xlsChart1

34034.1639833641461.52398336

4070

4-10

40-18

340238

The Quartic vs. the Approximate Factorization

Sheet1

Finding the Complex Zeros

of Quartic Functions


complex zeros

of fourth degree polynomials

y = x4 + ax3 + bx2 + cx + d

that have no real zeros.


a =06250


b =562100

Trace along the quartic curve

What is the value for c (-5 to 5):x0 =-1.8853

c =06250then y0 =34.16

What is the value for d (-25 to 25):The factorization of the quartic is roughly

d =462540y = (x^2 - 2.00x - 1.00 ) (x^2 + 2.00x + 10.00)

The zeros of the two quadratic factors are roughly

1st :x = 2.41 and x = -0.41

For a possible quadratic factor of the form x2 + ax + b:2nd :x = -1.00 + 3.00i and x = -1.00 - 3.00i

What is the value for a (-5 to 5):30-2

What is the value for b (-5 to 5):40-1

Click each item below for suggestions and investigations

Item 1

Item 2

Item 3





under grants DUE-0310123 and DUE-0442160.The Sum of the Squares of

the differences in heights is563432.48

Sheet1

34034.16398336340

4040

44

4040

340340

Sheet2

34034.1639833641461.52398336

4070

4-10

40-18

340238


Sheet3

plotting points:xyapproxSum of the Squares

-4340414-4340414-74.005476.0000

-24070-3.96328.32401.44-73.125346.5344

Disc =804-10-3.92316.96389.20-72.245218.6176

240-18-3.88305.91377.27-71.365092.2496

D > 04340238-3.84295.16365.64-70.484967.4304

root1 =2.414-3.8284.71354.31-69.604844.1600

root2 =-0.414-1.8834.1661.52-3.76274.56343.28-68.724722.4384

-3.72264.69332.53-67.844602.2656

Second Quadratic factor:-3.68255.11322.07-66.964483.6416

D = 0a =1-3.64245.80311.88-66.084366.5664

root3 =0.000b =2-3.6236.76301.96-65.204251.0400

root4 =0.000c =10-3.56227.99292.31-64.324137.0624

-3.52219.47282.91-63.444024.6336

-3.48211.21273.77-62.563913.7536

D < 0-3.44203.20264.88-61.683804.4224

root5 =10i-3.4195.43256.23-60.803696.6400

root6 =10i-3.36187.90247.82-59.923590.4064

-3.32180.61239.65-59.043485.7216

-3.28173.54231.70-58.163382.5856

-3.24166.69223.97-57.283280.9984

-3.2160.06216.46-56.403180.9600

Disc2 =-36-3.16153.64209.16-55.523082.4704

-3.12147.43202.07-54.642985.5296

D > 0-3.08141.42195.18-53.762890.1376

root1 =0.000-3.04135.62188.50-52.882796.2944

root2 =0.000-3130.00182.00-52.002704.0000

-2.96124.57175.69-51.122613.2544

-2.92119.33169.57-50.242524.0576

D = 0-2.88114.27163.63-49.362436.4096

root3 =0.000-2.84109.38157.86-48.482350.3104

root4 =0.000-2.8104.67152.27-47.602265.7600

-2.76100.12146.84-46.722182.7584

-2.7295.73141.57-45.842101.3056

D < 0-2.6891.50136.46-44.962021.4016

root5 =-13i-2.6487.42131.50-44.081943.0464

root6 =-1-3i-2.683.50126.70-43.201866.2400

-2.5679.72122.04-42.321790.9824

-2.5276.08117.52-41.441717.2736

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-2.4469.21108.89-39.681574.5024

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-2.3662.87100.79-37.921437.9264

-2.3259.8896.92-37.041371.9616

-2.2857.0293.18-36.161307.5456

-2.2454.2689.54-35.281244.6784

-2.251.6386.03-34.401183.3600

-2.1649.1082.62-33.521123.5904

-2.1246.6779.31-32.641065.3696

-2.0844.3576.11-31.761008.6976

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-240.0070.00-30.00900.0000

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-1.8834.1661.52-27.36748.5696

-1.8432.3958.87-26.48701.1904

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-1.6424.6846.76-22.08487.5264

-1.623.3544.55-21.20449.4400

-1.5622.0942.41-20.32412.9024

-1.5220.8940.33-19.44377.9136

-1.4819.7538.31-18.56344.4736

-1.4418.6736.35-17.68312.5824

-1.417.6434.44-16.80282.2400

-1.3616.6732.59-15.92253.4464

-1.3215.7530.79-15.04226.2016

-1.2814.8829.04-14.16200.5056

-1.2414.0527.33-13.28176.3584

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-110.0018.00-8.0064.0000

-0.969.4616.58-7.1250.6944

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-0.87.6111.21-3.6012.9600

-0.767.229.94-2.727.3984

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-0.525.432.872.566.5536

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-0.445.010.694.3218.6624

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-0.124.07-7.2911.36129.0496

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04.00-10.0014.00196.0000

0.044.01-10.8714.88221.4144

0.084.03-11.7315.76248.3776

0.124.07-12.5716.64276.8896

0.164.13-13.3917.52306.9504

0.24.20-14.2018.40338.5600

0.244.29-14.9919.28371.7184

0.284.40-15.7620.16406.4256

0.324.52-16.5221.04442.6816

0.364.66-17.2621.92480.4864

0.44.83-17.9722.80519.8400

0.445.01-18.6723.68560.7424

0.485.21-19.3524.56603.1936

0.525.43-20.0125.44647.1936

0.565.67-20.6526.32692.7424

0.65.93-21.2727.20739.8400

0.646.22-21.8628.08788.4864

0.686.53-22.4328.96838.6816

0.726.86-22.9829.84890.4256

0.767.22-23.5030.72943.7184

0.87.61-23.9931.60998.5600

0.848.03-24.4532.481054.9504

0.888.47-24.8933.361112.8896

0.928.95-25.2934.241172.3776

0.969.46-25.6635.121233.4144

110.00-26.0036.001296.0000

1.0410.58-26.3036.881360.1344

1.0811.19-26.5737.761425.8176

1.1211.85-26.7938.641493.0496

1.1612.54-26.9839.521561.8304

1.213.27-27.1340.401632.1600

1.2414.05-27.2341.281704.0384

1.2814.88-27.2842.161777.4656

1.3215.75-27.2943.041852.4416

1.3616.67-27.2543.921928.9664

1.417.64-27.1644.802007.0400

1.4418.67-27.0145.682086.6624

1.4819.75-26.8146.562167.8336

1.5220.89-26.5547.442250.5536

1.5622.09-26.2348.322334.8224

1.623.35-25.8549.202420.6400

1.6424.68-25.4050.082508.0064

1.6826.08-24.8850.962596.9216

1.7227.54-24.3051.842687.3856

1.7629.08-23.6452.722779.3984

1.830.70-22.9053.602872.9600

1.8432.39-22.0954.482968.0704

1.8834.16-21.2055.363064.7296

1.9236.02-20.2256.243162.9376

1.9637.97-19.1557.123262.6944

240.00-18.0058.003364.0000

2.0442.13-16.7558.883466.8544

2.0844.35-15.4159.763571.2576

2.1246.67-13.9760.643677.2096

2.1649.10-12.4261.523784.7104

2.251.63-10.7762.403893.7600

2.2454.26-9.0263.284004.3584

2.2857.02-7.1464.164116.5056

2.3259.88-5.1665.044230.2016

2.3662.87-3.0565.924345.4464

2.465.98-0.8266.804462.2400

2.4469.211.5367.684580.5824

2.4872.584.0268.564700.4736

2.5276.086.6469.444821.9136

2.5679.729.4070.324944.9024

2.683.5012.3071.205069.4400

2.6487.4215.3472.085195.5264

2.6891.5018.5472.965323.1616

2.7295.7321.8973.845452.3456

2.76100.1225.4074.725583.0784

2.8104.6729.0775.605715.3600

2.84109.3832.9076.485849.1904

2.88114.2736.9177.365984.5696

2.92119.3341.0978.246121.4976

2.96124.5745.4579.126259.9744

3130.0050.0080.006400.0000

3.04135.6254.7480.886541.5744

3.08141.4259.6681.766684.6976

3.12147.4364.7982.646829.3696

3.16153.6470.1283.526975.5904

3.2160.0675.6684.407123.3600

3.24166.6981.4185.287272.6784

3.28173.5487.3886.167423.5456

3.32180.6193.5787.047575.9616

3.36187.9099.9887.927729.9264

3.4195.43106.6388.807885.4400

3.44203.20113.5289.688042.5024

3.48211.21120.6590.568201.1136

3.52219.47128.0391.448361.2736

3.56227.99135.6792.328522.9824

3.6236.76143.5693.208686.2400

3.64245.80151.7294.088851.0464

3.68255.11160.1594.969017.4016

3.72264.69168.8595.849185.3056

3.76274.56177.8496.729354.7584

3.8284.71187.1197.609525.7600

3.84295.16196.6898.489698.3104

3.88305.91206.5599.369872.4096

3.92316.96216.72100.2410048.0576

3.96328.32227.20101.1210225.2544

4340.00238.00102.0010404.0000

Sum =563432.48

1. Every quartic polynomial has precisely four zeros. Since complex zeros occur only as complex conjugate pairs, a quartic can have either four real zeros, two real zero and a pair of complex zeros, or two pairs of complex zeros. This DIGMath spreadsheet lets you approximate all four zeros of any quartic of the form y = x4 + ax3 + bx2 + cx + d through a combination of graphical investigation and algebraic computation that takes place behind the scenes. Start by defining your quartic polynomial using the sliders above to select values for the four coefficients a, b, c, and d. Then use the other two sliders above to estimate the coefficients of an approximate quadratic factor of the form x2 + ax + b. This quadratic is then divided into the original fourth degree polynomial to produce a second quadratic plus a remainder. The remainder term is ignored, leaving a product of two approximate quadratic factors.

2. The spreadsheet displays the graph of the original quartic, along with a tracing point. It also shows the approximate factorization of the quartic along with the four zeros, either real or complex. the spreadsheet also shows in the lower chart, the graphs of the original quartic and the product of the two approximate factors. Finally, it shows the value for the Sum of the Squares of the vertical distances between the two curves to measure how close the curves are to each other. The idea is to vary the coefficients a and b to produce the best match between the quartic and the approximate factorizations, as determined by the closer the two curves are to one another and the smaller the Sum of the Squares is. If you can achieve a perfect match, as indicated by a value of 0 for the Sum of the Squares, you will have found the exact factorization of the quartic and hence the correct values for the four zeros, correct to two decimal places.

3. We suggest that you start with a quartic whose factors you know, most likely by starting with either four linear factors and multiplying them out or with two linear factors and one quadratic factor and multiplying them out, or two different quadratic factors and multiplying them out. For example, you might try (x2 + 1)(x2 + 4) or (x2 + 2x + 2)(x2 - 2x + 2). This will give you the chance to see how the spreadsheet operates. Then feel free to vary the values of the parameters as much as you want and watch how the two curves in the lower chart approach each other and the Sum of the Squares approaches 0 while the two quadratic approximations approach the actual two quadratic factors and the zeros of the approximate quadratic factors approach those of the actual quadratic factor.

MBD00000050.unknown

MBD000000F0.unknown

MBD01718BCD.unknown

MBD017167DC.unknown

MBD000000A0.unknown

MBD000000C8.unknown

MBD00000078.unknown

MBD00000028.unknown

_1472540289.xlsChart2

34034.1639833634236.16398336

4042

46

4042

340342


Sheet1

Finding the Complex Zeros

of Quartic Functions


complex zeros

of fourth degree polynomials

y = x4 + ax3 + bx2 + cx + d

that have no real zeros.


a =06250


b =562100

Trace along the quartic curve

What is the value for c (-5 to 5):x0 =-1.8853

c =06250then y0 =34.16

What is the value for d (-25 to 25):The factorization of the quartic is roughly

d =462540y = (x^2 + 0.00x + 2.00 ) (x^2 + 0.00x + 3.00)

The zeros of the two quadratic factors are roughly

1st :x = 0.00 + 1.41i and x = 0.00 - 1.41i

For a possible quadratic factor of the form x2 + ax + b:2nd :x = 0.00 + 1.73i and x = 0.00 - 1.73i

What is the value for a (-5 to 5):500

What is the value for b (-5 to 5):702

Click each item below for suggestions and investigations

Item 1

Item 2

Item 3





under grants DUE-0310123 and DUE-0442160.The Sum of the Squares of

the differences in heights is804.00

Sheet1

34034.16398336340

4040

44

4040

340340

Sheet2

34034.1639833634236.16398336

4042

46

4042

340342


Sheet3

plotting points:xyapproxSum of the Squares

-4340342-4340342-2.004.0000

-24042-3.96328.32330.32-2.004.0000

Disc =-8046-3.92316.96318.96-2.004.0000

24042-3.88305.91307.91-2.004.0000

D > 04340342-3.84295.16297.16-2.004.0000

root1 =0.000-3.8284.71286.71-2.004.0000

root2 =0.000-1.8834.1636.16-3.76274.56276.56-2.004.0000

-3.72264.69266.69-2.004.0000

Second Quadratic factor:-3.68255.11257.11-2.004.0000

D = 0a =1-3.64245.80247.80-2.004.0000

root3 =0.000b =0-3.6236.76238.76-2.004.0000

root4 =0.000c =3-3.56227.99229.99-2.004.0000

-3.52219.47221.47-2.004.0000

-3.48211.21213.21-2.004.0000

D < 0-3.44203.20205.20-2.004.0000

root5 =01.4142135624i-3.4195.43197.43-2.004.0000

root6 =0-1.4142135624i-3.36187.90189.90-2.004.0000

-3.32180.61182.61-2.004.0000

-3.28173.54175.54-2.004.0000

-3.24166.69168.69-2.004.0000

-3.2160.06162.06-2.004.0000

Disc2 =-12-3.16153.64155.64-2.004.0000

-3.12147.43149.43-2.004.0000

D > 0-3.08141.42143.42-2.004.0000

root1 =0.000-3.04135.62137.62-2.004.0000

root2 =0.000-3130.00132.00-2.004.0000

-2.96124.57126.57-2.004.0000

-2.92119.33121.33-2.004.0000

D = 0-2.88114.27116.27-2.004.0000

root3 =0.000-2.84109.38111.38-2.004.0000

root4 =0.000-2.8104.67106.67-2.004.0000

-2.76100.12102.12-2.004.0000

-2.7295.7397.73-2.004.0000

D < 0-2.6891.5093.50-2.004.0000

root5 =01.7320508076i-2.6487.4289.42-2.004.0000

root6 =0-1.7320508076i-2.683.5085.50-2.004.0000

-2.5679.7281.72-2.004.0000

-2.5276.0878.08-2.004.0000

-2.4872.5874.58-2.004.0000

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110.0012.00-2.004.0000

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1.1612.5414.54-2.004.0000

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1.8834.1636.16-2.004.0000

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240.0042.00-2.004.0000

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2.1246.6748.67-2.004.0000

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2.88114.27116.27-2.004.0000

2.92119.33121.33-2.004.0000

2.96124.57126.57-2.004.0000

3130.00132.00-2.004.0000

3.04135.62137.62-2.004.0000

3.08141.42143.42-2.004.0000

3.12147.43149.43-2.004.0000

3.16153.64155.64-2.004.0000

3.2160.06162.06-2.004.0000

3.24166.69168.69-2.004.0000

3.28173.54175.54-2.004.0000

3.32180.61182.61-2.004.0000

3.36187.90189.90-2.004.0000

3.4195.43197.43-2.004.0000

3.44203.20205.20-2.004.0000

3.48211.21213.21-2.004.0000

3.52219.47221.47-2.004.0000

3.56227.99229.99-2.004.0000

3.6236.76238.76-2.004.0000

3.64245.80247.80-2.004.0000

3.68255.11257.11-2.004.0000

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3.88305.91307.91-2.004.0000

3.92316.96318.96-2.004.0000

3.96328.32330.32-2.004.0000

4340.00342.00-2.004.0000

Sum =804.00

1. Every quartic polynomial has precisely four zeros. Since complex zeros occur only as complex conjugate pairs, a quartic can have either four real zeros, two real zero and a pair of complex zeros, or two pairs of complex zeros. This DIGMath spreadsheet lets you approximate all four zeros of any quartic of the form y = x4 + ax3 + bx2 + cx + d through a combination of graphical investigation and algebraic computation that takes place behind the scenes. Start by defining your quartic polynomial using the sliders above to select values for the four coefficients a, b, c, and d. Then use the other two sliders above to estimate the coefficients of an approximate quadratic factor of the form x2 + ax + b. This quadratic is then divided into the original fourth degree polynomial to produce a second quadratic plus a remainder. The remainder term is ignored, leaving a product of two approximate quadratic factors.

2. The spreadsheet displays the graph of the original quartic, along with a tracing point. It also shows the approximate factorization of the quartic along with the four zeros, either real or complex. the spreadsheet also shows in the lower chart, the graphs of the original quartic and the product of the two approximate factors. Finally, it shows the value for the Sum of the Squares of the vertical distances between the two curves to measure how close the curves are to each other. The idea is to vary the coefficients a and b to produce the best match between the quartic and the approximate factorizations, as determined by the closer the two curves are to one another and the smaller the Sum of the Squares is. If you can achieve a perfect match, as indicated by a value of 0 for the Sum of the Squares, you will have found the exact factorization of the quartic and hence the correct values for the four zeros, correct to two decimal places.

3. We suggest that you start with a quartic whose factors you know, most likely by starting with either four linear factors and multiplying them out or with two linear factors and one quadratic factor and multiplying them out, or two different quadratic factors and multiplying them out. For example, you might try (x2 + 1)(x2 + 4) or (x2 + 2x + 2)(x2 - 2x + 2). This will give you the chance to see how the spreadsheet operates. Then feel free to vary the values of the parameters as much as you want and watch how the two curves in the lower chart approach each other and the Sum of the Squares approaches 0 while the two quadratic approximations approach the actual two quadratic factors and the zeros of the approximate quadratic factors approach those of the actual quadratic factor.

MBD00000050.unknown

MBD000000F0.unknown

MBD0171B951.unknown

MBD017167DC.unknown

MBD000000A0.unknown

MBD000000C8.unknown

MBD00000078.unknown

MBD00000028.unknown

_1472539464.unknown

_1472538405.xlsChart1

-1341-135

-32-33

-2-3

43

3433


Sheet1

Finding the Zeros

of Cubic Functions


three zeros of any cubic polynomial

y = ax3 + bx2 + cx + d

whether they are all real zeros or

one real and two complex zeros.


a =160


b =-320

Trace along the curve to estimate a real zero

What is the value for c (-5 to 5):x0 =1500

c =5100

then y0 =1.00

What is the value for d (-5 to 5):

d =-230The factorization of the cubic is roughly

y = (x - 1.00) (1.00x^2 - 2.00x + 3.00)

The zeros of the quadratic factor are roughly

Click each item below for suggestions and investigationsx = 1.00 + 1.41i and x = 1.00 - 1.41i

Item 1

Item 2

Item 3






1. Every cubic polynomial y = ax3 + bx2 + cx + d has precisely three zeros. Since complex zeros occur only as complex conjugate pairs, a cubic can have either three real zeros or one real zero and a pair of complex zeros. This DIGMath spreadsheet lets you find all three zeros of any cubic through a combination of graphical investigation and algebraic computation that takes place behind the scenes. Start by defining your cubic polynomial using the sliders above to select values for the four coefficients. Then use the slider to the right to trace along the cubic curve. Each point x0 is used as an approximation to a real zero of the function and the quotient of the function divided by the linear factor (x - x0) is displayed, ignoring the remainder, as an approximation to the factored form of the function.

2. The closer that x0 is to an actual real zero of the cubic, the more accurate the factorization is. In turn, the closer you are to a real zero, the closer that the zeros of the quadratic factor are to the actual zeros, either real or complex, of the original cubic polynomial.

3. We suggest that you start with a cubic that you know factors, most likely by starting with three linear factors and multiplying them out or with one linear factor and one quadratic factor and multiplying them out. This will give you the chance to see how the spreadsheet operates. Then feel free to vary the values of the parameters as much as you want and watch how the two curves in the lower chart approach each other while the quadratic term approaches the actual quadratic factor and the zeros of the approximate quadratic factor approach those of the actual quadratic factor.

Sheet1

-134-8.432

-32

-2

4

34

Sheet2

-1341-135

-32-33

-2-3

43

3433


Sheet3

plotting points:xyapprox00

-4-134-13500

-2-32-33

Disc =-80-2-3

243

D > 043433

root1 =0.000

root2 =0.00011

Quadratic factor:

D = 0a =1

root3 =0.000b =-2

root4 =0.000c =3

D < 0

root5 =11.4142135624i

root6 =1-1.4142135624i

MBD00000050.unknown

MBD000000A0.unknown

MBD016FEF5B.unknown

MBD016FEEB8.unknown

MBD00000078.unknown

MBD00000028.unknown

_1472457578.unknown

_1472536493.xlsChart2

34.10.8774011393.36191414354.55193679694.9480884861

10.3-2.2722981777-0.27003924140.68035761850.9686561603

74.52.57800250574.09800737374.808778444.9892238344

n = 0 n = 1 n = 2 n = 3 n = 4 n = 5

Sheet1

Bairstow's Method

for Finding the Complex Roots

of a Polynomial

This DIGMath illustrates and applies

Bairstow's Method for finding the complex roots

of any desired polynomial (up to 8th degree)

y = c8 x8+c7 x7+c6 x6+c5 x5+c4 x4+c3 x3+c2 x2+c1 x+c0

where c0, c1, c2, c3, c4, c5, c6, c7, c8 are any coefficients

on any interval from x = xMin to x = xMax.

First, select the degree ( k = 3 to 8) of your polynomial:

44

Enter an initial guess for a quadratic factor: y = x2 - A x - B

Next, enter the value for each of the coefficients in

y = c0 + c1x + c2x^2 + c3x^3+ c4x^4First,A =-1

(Use 0's if you don't want some of the terms.)

ThenB =-1

c0 =4

c1 =0

c2 =5The successive Bairstow quadratic approximations are:

c3 =0n = 0:Q(x) = x^2 + 1x + 1

c4 =14

015n = 1:Q(x) = x^2 + 0.1000x + 0.3000

026

007n = 2:Q(x) = x^2 + 0.4252x - 2.2723

028

n = 3:Q(x) = x^2 + 0.1840x - 0.2700

The graph of y = 4.00 + 0.00x + 5.00x^2 + 0.00x^3 + 1.00x^4

n = 4:Q(x) = x^2 + 0.0642x + 0.6804

Now enter the interval for x:

xMin =-2n = 5:Q(x) = x^2 + 0.0103x + 0.9687

xMax =2

The two complex zeros of the last quadratic are roughly:

x =-0.0321052054+0.9841898805i

Click each item below for suggestions and investigationsx =-0.0321052054-0.9841898805i

Item 1

Item 2

Item 3






Sheet1

34.10.8774011393.36191414354.55193679694.9480884861

10.3-2.2722981777-0.27003924140.68035761850.9686561603

74.52.57800250574.09800737374.808778444.9892238344

n = 0 n = 1 n = 2 n = 3 n = 4 n = 5

Sheet2

a00 - i00a0 - i0deg n = 3Bairstow 1bcDenom'sNext uNext vbcDenom'sNext uNext vbcDenom'sNext uNext vbcDenom'sNext uNext vbcDenom'sNext uNext v

xy4101111111111

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-1.8432.39028736


-1.830.6976Delta x=0.0201111111111

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-1.7227.54413056iteration43520-0.1-0.32.63111.3845-4.39-0.42515034172.272298177722.693438404350.016340346392.9395192964-0.18402330760.27003924145.62102139197.542595882531.0322621828-0.0642104108-0.68035761851.0733149838-1.386648254713.2856403543-0.0102838371-0.9686561603

-1.726.8021iu(i)v(i)51434-0.1176470588-0.3235294118-0.13081-0.0220619.4601-0.0183694791-0.1183477726-19.0434750833-63.8382730845-397.2752833746-0.5331576642-0.0883868257-1.3113877681-3.2835829469-37.8728930723-0.3097506271-0.77451099090.09024806810.43853342035.20875589010.0775773701-0.9594879216

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-1.6424.681948162-0.42515034172.2722981777812121-0.6446280992-0.59504132230.221187910.553763291.6472012521-0.0185755495-0.1230871452164.7526952635747.333877301221099.7855891785-0.2715306411.67948587260.59980183262.224977752311.5661643112-0.0964152530.18100097540.50167351970.40054081840.17420993911.2131842681-1.170175675

-1.6224.009475363-0.18402330760.2700392414

-1.623.35364-0.0642104108-0.6803576185deg n = 5Bairstow 3bcDenom'sNext uNext vbcDenom'sNext uNext vbcDenom'sNext uNext vbcDenom'sNext uNext vbcDenom'sNext uNext v

-1.5822.714012965-0.0102838371-0.968656160301111111111

-1.5622.090408961-1-2-0.1-0.2-0.4251503417-0.8503006834-0.1840233076-0.3680466151-0.0642104108-0.1284208216

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-1.5220.889948163-4-8-0.441-0.824-4.134735522-10.3553019777-1.0257354382-2.1633216484-0.2339447076-0.381046807-3.6351617862-0.1526017427-5.0154742643

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-1.4418.66781696731281-0.6172839506-0.50617283950.11686790.24242521.89866281-0.02379923710.1876290976-68.6380279387-298.92134440811840.569706111-0.22101012396.3894815307-0.6778640584-2.3743134538-22.4075786159-0.21364264270.0239064988-0.0141399864-0.32400767650.44083415080.7124585663-0.965532743

-1.4218.14786896812121-0.6446280992-0.59504132230.221187910.553763291.6472012521-0.0185755495-0.1230871452164.7526952635747.3338773012-5950.7254672465-0.96984741524.37426248450.59980183262.224977752311.5661643112-0.0964152530.18100097540.50167351970.40054081840.17420993911.2131842681-1.170175675

-1.417.6416


-1.3616.6690201601111111111

-1.3416.202179361-1-2-0.1-0.2-0.4251503417-0.8503006834-0.1840233076-0.3680466151-0.0642104108-0.1284208216

-1.3215.747957762564.714.437.453050990710.08685479455.30390381915.6416722164.32376535843.6516536936

-1.315.30613-4-8-0.441-0.824-4.134735522-10.3553019777-1.0257354382-2.1633216484-0.2339447076-0.381046807-3.6351617862-0.1526017427-5.0154742643

-1.2814.876354564352.63111.384522.693438404350.01634034635.62102139197.54259588251.0733149838-1.386648254713.2856403543-0.0102838371-0.9686561603

-1.2614.45847376514-0.13081-0.02206-19.0434750833-63.8382730845-1.3113877681-3.28358294690.09024806810.43853342035.20875589010.0775773701-0.9594879216

-1.2414.052213766-4-1357-0.5263157895-0.5789473684-0.776249-1.189393-5.454359-0.2504733409-0.497567195259.6625986698200.4555013071-397.2752833746-1.26754028936.72362538621.75922226724.400274932149.7873065728-0.06179326980.0900123885-0.73603289170.17922540142.08989514180.1298688525-1.149779223

-1.2213.65733456731281-0.6172839506-0.50617283950.11686790.24242521.89866281-0.02379923710.1876290976-68.6380279387-298.92134440811840.569706111-0.22101012396.3894815307-0.6778640584-2.3743134538-22.4075786159-0.21364264270.0239064988-0.0141399864-0.32400767650.44083415080.7124585663-0.965532743

-1.213.2736812121-0.6446280992-0.59504132230.221187910.553763291.6472012521-0.0185755495-0.1230871452164.7526952635747.3338773012-5950.7254672465-0.96984741524.37426248450.59980183262.224977752311.5661643112-0.0964152530.18100097540.50167351970.40054081840.17420993911.2131842681-1.170175675

-1.1812.90077776


-1.1412.1869601601111111111

-1.1211.845519361-1-2-0.1-0.2-0.4251503417-0.8503006834-0.1840233076-0.3680466151-0.0642104108-0.1284208216

-1.111.51412564.714.437.453050990710.08685479455.30390381915.6416722164.32376535843.6516536936

-1.0811.192488963-4-8-0.441-0.824-4.134735522-10.3553019777-1.0257354382-2.1633216484-0.2339447076-0.381046807-3.6351617862-0.1526017427-5.0154742643

-1.0610.880476964352.63111.384522.693438404350.01634034635.62102139197.54259588251.0733149838-1.386648254713.2856403543-0.0102838371-0.9686561603

-1.0410.57785856514-0.13081-0.02206-19.0434750833-63.8382730845-1.3113877681-3.28358294690.09024806810.43853342035.20875589010.0775773701-0.9594879216

-1.0210.284432166-4-13-0.776249-1.18939359.6625986698200.45550130711.75922226724.4002749321-0.73603289170.17922540142.08989514180.1298688525-1.149779223

-110731281-0.6172839506-0.50617283950.11686790.24242521.89866281-0.02379923710.1876290976-68.6380279387-298.92134440811840.569706111-0.22101012396.3894815307-0.6778640584-2.3743134538-22.4075786159-0.21364264270.0239064988-0.0141399864-0.32400767650.44083415080.7124585663-0.965532743

-0.989.72436816812121-0.6446280992-0.59504132230.221187910.553763291.6472012521-0.0185755495-0.1230871452164.7526952635747.3338773012-5950.7254672465-0.96984741524.37426248450.59980183262.224977752311.5661643112-0.0964152530.18100097540.50167351970.40054081840.17420993911.2131842681-1.170175675

-0.969.45734656


-0.928.94839296

finding the complex roots of polynomials

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complex roots of polynomials

complex zeros of polynomials

real roots

roots of polynomial

complex zeros of cubics

polynomials of degree

cubic polynomial

direct way