How do I solve two polynomials

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Determine zeros (solutions) of polynomials

Dynamic explanation of the solution procedure for polynomials 2.-4. Degree
Numerical determination of the complex zeros for polynomials of higher degree


Hints

You can enter polynomials or equations that lead to a polynomial above or the coefficients of a polynomial 2.-4. Degree directly into the input fields for the corresponding polynomial degrees. In the first option, the program tries to convert the equation / term into the standard form anxn + an-1xn-1 + ... + a1x + a0 = Bring 0. If the input does not contain any fractions or brackets, a symbolic attempt is made first, otherwise or if the first method fails, a system of linear equations is used. To do this, the maximum degree of the polynomial must be specified (at most 25). If the algorithm does not find the polynomial right away, the process can and should be repeated; the program works with random numbers for the x-values; in addition, high potencies often result in large overall errors in the floating point calculation.

If fraction equations / terms are to be resolved into polynomials by multiplication with the denominators, the corresponding option must be activated. (The LCM of the denominator is not determined, and it is not shortened beforehand.)

If a polynomial up to the 4th degree is found, the coefficients are entered in the input fields of the corresponding polynomial and an explanation of the solution procedure can be generated with the button [Solve with explanation].

As an alternative to the Javascript mode, a Java applet integrated into the page can also be used to search for zeros. The applet is optimized for very fast searches - it should serve to dynamically visualize the zeros in the Gaussian number plane in "real time", while the polynomial coefficients are simply changed with scroll bars - but sometimes fails.

Samples are made, with the complex zeros exclusively with the (possibly calculated) standardized polynomial, with the real ones also with the entered equation or the entered term. With equations (), the value is calculated.

The Javascript method is used for the test with the entered equation, whereby powers are converted into the Javascript syntax beforehand: e.g. for x = 4.789 in. Polynomials are always calculated using the Horner scheme, which requires considerably fewer multiplications and also works in complex areas. In addition to considerable speed advantages, this method is also much more accurate than (due to the smaller number of floating point multiplications required). This is shown, for example, by the sample with the real zero found by the script x = 1.9999999701976665 of the polynomial x ^ 25 - x ^ 24 - x ^ 23 - x ^ 22 - x ^ 21 - x ^ 20 - x ^ 19 - x ^ 18 - x ^ 17 - x ^ 16 - x ^ 15 - x ^ 14 - x ^ 13 - x ^ 12 - x ^ 11 - x ^ 10 - x ^ 9 - x ^ 8 - x ^ 7 - x ^ 6 - x ^ 5 - x ^ 4 - x ^ 3 - x ^ 2 - x - 1. results in the (completely wrong) value -1021, thus suggests that this zero is wrong. The Horner algorithm calculates (relatively correctly) the value 6.616929226765933e-14, which is very close to zero. In fact, all 16 digits of the zero are correct.

Real zeros and conjugate complex zero pairs usually lead to polynomial division in the program, in which the polynomial is simplified, i.e. its degree is reduced. The log of the polynomial divisions is displayed in the results window under the sample. In the case of (conjugate) complex zeros, division by. The solution methods for cubic and biquadratic polynomials have not yet been implemented in the numerical search algorithm.

 

I have tried to take into account as many special cases as possible in the automatic creation of the explanations for the solution procedures, so that the solution path can be easily understood for almost all cases. But that had its limits, both due to the limited display options in the text field and the desire not to let the script get too big. The equations of the form xn = y or other cases in which the absolute term is missing are not dealt with separately, neither in the explanations nor in the calculation itself.

Equations in which the absolute term is missing, such as x4 + 4x3 - 2x = 0, can be better controlled by factoring out an x: x4 + 4x3 - 2x = x (x3 + 4x2 - 2) = 0.
The resulting product becomes zero when at least one factor is zero; i.e. the solutions are obtained by considering the factors separately or by solving the equations x = 0 and x3 + 4x2 - 2 = 0.

There are no general solution formulas for polynomials of higher order. The main theorem of algebra says, however, that polynomials of degree n always have exactly n (possibly complex) zeros, but not all of them have to be different.
If one or more real zeros are found by guessing, trial and error, by reading the graph (→ function plotter) or by numerical methods (e.g. the Newton method described briefly above), the polynomial can be divided by polynomial division by the term (x-x0) into a polynomial that is one degree smaller and contains the remaining zeros. x0 stands for the x-value of the zero.
Example: The polynomial x6 - 4x5 + 5x4 - 13x2 + 25x - 14 = 0 has zeros at x = 1 and x = 2, as can be found out quite easily using one of the methods mentioned.
The polynomial divisions then result in: (x6-4x5+ 5x4-13x2+ 25x-14) / (x-1) = x5-3x4+ 2x3+ 2x2-11x + 14 and (x5-3x4+ 2x3+ 2x2-11x + 14) / (x-2) = x4-x3+ 2x-7. The zeros of this polynomial can then be obtained using the solution formula described above.

→ Page on polynomial division
→ Page for numerical solving of equations

How "good" the results are can be seen from the automatic test. The solutions of the last calculation in each case were inserted into the given polynomial, and you can see here how close the result is to zero. Note that small values ​​are listed in exponential notation, the specification 3.3306690738754696e-16 + 2.704619853023893e-16 · î means approximately: 0.0000000000000003330669 ... + 0.00000000000000027046 ... î. So that's closer to zero than normal pocket calculators can record. The abundance of root and occasional cosine calculations, which in the best computers can only be carried out numerically using approximation methods, creates a natural inaccuracy. As a rule, however, the springs that have been left can be glued back on by applying a suitable method for approximating zeros, such as the Newton algorithm, which is implemented in this script, to the results. It also works with complex zeros.


Version: May 22, 2004
© Arndt Brünner
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Applet for displaying the zeros in the numerical level
Real number calculator
Complex number calculator
Exercises in solving quadratic equations

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