A0, A1, A2, A3 ... An are the polynomial coefficients
An nth degree polynomial has n+1 coefficients. The polynomial roots or zeros are the ϰ values where the polynomial value is zero. An nth degree polymonial has n roots or zeros.
(ϰ - C0) (ϰ - C1) (ϰ - C1).... (ϰ - Cn-1) = 0
C0, C1, C2... Cn-1 are the roots or zeros of the polynomial
The polynomial maxima or minima are the ϰ values where the slope is zero. An nth degree polymonial has n-1 maximum or minimum points.
The polynomial inflection points are the ϰ values where the curvature is zero. An nth degree polymonial has n-2 inflection points.
For polynomials with two or more complex roots use the single root calculator with an initial guess value to find single real roots. Use a plot page to plot the polynomial and identify the approximate root values if any.
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| CALC : Math : Polynomial 001 : First Degree Linear Polynomial Real Root Or Zero : Calculator
Description : Calculate the root or zero of a linear first order polynomial.
Discussion : A first order polynomial is a straight line. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and check the polynomial values. The function PolyRoots() calculates the polynomial roots.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Roots Or Zeros
- P = Output Value
Calculation :
C = PolyRoots( A )
P = A1 ϰ + A0
δP = A1
δ²P = 0
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| CALC : Math : Polynomial 002 : Second Degree Quadratic Polynomial Real Roots Or Zeros : Calculator
Description : Calculate the real roots or zeros of a quadratic second order polynomial.
Discussion : The tool calculates the roots (value equals zero), and the maximum or minimum point (slope equals zero). All roots of the polynomial must be real. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and to check the values. The functions PolyRoots() and PolyMaxMins() calculate the polynomial roots and the polynomial maxima minima respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Roots Or Zeros
- M = Maxima Or Minima
- P = Output Value
Calculation :
C = PolyRoots( A )
M = PolyMaxMins( A )
P = ( A2 ϰ + A1 ) ϰ + A0
δP = 2 A2 ϰ + A1
δ²P = 2 A2
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| CALC : Math : Polynomial 003 : Third Degree Cubic Polynomial Real Roots Or Zeros : Calculator
Description : Calculate the real roots or zeros of a cubic third order polynomial.
Discussion : The tool calculates the roots (value equals zero), the maximum or minimum points (slope equals zero) and the inflection point (curvature equals zero). All roots of the polynomial must be real. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and to check the values. The roots are calculated using the Durand Kerner method. The functions PolyRoots(), PolyMaxMins() and PolyInflections() calculate the polynomial roots, the polynomial maxima minima and the polynomial inflection points respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Roots Or Zeros
- I = Inflection Points
- M = Maxima Or Minima
- P = Output Value
Calculation :
C = PolyRoots( A )
M = PolyMaxMins( A )
I = PolyInflections( A )
P = ( ( A3 ϰ + A2 ) ϰ + A1 ) ϰ + A0
δP = ( 3 A3 ϰ + 2 A2 ) ϰ + A1
δ²P = 6 A3 ϰ + 2 A2
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| CALC : Math : Polynomial 004 : Fourth Degree Quartic Polynomial Real Roots Or Zeros : Calculator
Description : Calculate the real roots or zeros of a quartic fourth order polynomial.
Discussion : The tool calculates the roots (value equals zero), the maximum or minimum points (slope equals zero) and the inflection points (curvature equals zero). All roots of the polynomial must be real. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and to check the values. The roots are calculated using the Durand Kerner method. The functions PolyRoots(), PolyMaxMins() and PolyInflections() calculate the polynomial roots, the polynomial maxima minima and the polynomial inflection points respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Roots Or Zeros
- I = Inflection Points
- M = Maxima Or Minima
- P = Output Value
Calculation :
C = PolyRoots( A )
M = PolyMaxMins( A )
I = PolyInflections( A )
P = ( ( ( A4 ϰ + A3 ) ϰ + A2 ) ϰ + A1 ) ϰ + A0
δP = ( ( 4 A4 ϰ + 3 A3 ) ϰ + 2 A2 ) ϰ + A1
δ²P = ( 12 A4 ϰ + 6 A3 ) ϰ + 2 A2
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| CALC : Math : Polynomial 005 : Fifth Degree Quintic Polynomial Real Roots Or Zeros : Calculator
Description : Calculate the real roots or zeros of a quintic fifth order polynomial.
Discussion : The tool calculates the roots (value equals zero), the maximum or minimum points (slope equals zero) and the inflection points (curvature equals zero). All roots of the polynomial must be real. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and to check the values. The roots are calculated using the Durand Kerner method. The functions PolyRoots(), PolyMaxMins() and PolyInflections() calculate the polynomial roots, the polynomial maxima minima and the polynomial inflection points respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Roots Or Zeros
- I = Inflection Points
- M = Maxima Or Minima
- P = Output Value
Calculation :
C = PolyRoots( A )
M = PolyMaxMins( A )
I = PolyInflections( A )
P = ( ( ( ( A5 ϰ + A4 ) ϰ + A3 ) ϰ + A2 ) ϰ + A1 ) ϰ + A0
δP = ( ( ( 5 A5 ϰ + 4 A4 ) ϰ + 3 A3 ) ϰ + 2 A2 ) ϰ + A1
δ²P = ( ( 20 A5 ϰ + 12 A4 ) ϰ + 6 A3 ) ϰ + 2 A2
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| CALC : Math : Polynomial 006 : Nth Degree Polynomial Real Roots Or Zeros : Calculator
Description : Calculate the real roots or zeros of an nth order polynomial.
Discussion : Enter the polynomial coefficients as a comma separated list in order A0, A1, A2....An. The tool calculates the roots (value equals zero), the maximum or minimum points (slope equals zero) and the inflection points (curvature equals zero). All roots of the polynomial must be real. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and to check the values. The roots are calculated using the Durand Kerner method. The functions PolyRoots(), PolyMaxMins() and PolyInflections() calculate the polynomial roots, the polynomial maxima minima and the polynomial inflection points respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Roots Or Zeros
- I = Inflection Points
- M = Maxima Or Minima
- P = Output Value
Calculation :
P = PolyValue( A , ϰ )
δP = PolySlope( A , ϰ )
δ²P = PolyCurve( A , ϰ )
C = PolyRoots( A )
M = PolyMaxMins( A )
I = PolyInflections( A )
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| CALC : Math : Polynomial 012 : Second Degree Quadratic Polynomial Single Real Root Or Zero : Calculator
Description : Calculate a single real root or zero of a quadratic second order polynomial.
Discussion : The tool calculates the root (value equals zero), and the maximum or minimum point (slope equals zero). The guess value should be close to the root value and or the maximum minimum value. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and to estimate the guess values. The tool uses Newtons method to find the root. The functions PolyRoot() and PolyMaxMin() calculate the polynomial root and the polynomial maxima minima respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
- g = Initial Guess
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Root Or Zero
- M = Maxima Or Minima
- P = Output Value
Calculation :
C = PolyRoot( A , g )
M = PolyMaxMin( A , g )
P = ( A2 ϰ + A1 ) ϰ + A0
δP = 2 A2 ϰ + A1
δ²P = 2 A2
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| CALC : Math : Polynomial 013 : Third Degree Cubic Polynomial Single Real Root Or Zero : Calculator
Description : Calculate a single real root or zero of a cubic third order polynomial.
Discussion : The tool calculates the root (value equals zero), the maximum or minimum point (slope equals zero) and the inflection point (curvature equals zero). The guess value should be close to the root value, maximum minimum value or inflection value. The root of the polynomial must be real. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and to check the values. The tool uses Newtons method to find the root. The functions PolyRoot(), PolyMaxMin() and PolyInflection() calculate the polynomial roots, the polynomial maxima minima and the polynomial inflection points respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
- g = Initial Guess
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Root Or Zero
- I = Inflection Point
- M = Maxima Or Minima
- P = Output Value
Calculation :
C = PolyRoot( A , g )
M = PolyMaxMin( A , g )
I = PolyInflection( A , g )
P = ( ( A3 ϰ + A2 ) ϰ + A1 ) ϰ + A0
δP = ( 3 A3 ϰ + 2 A2 ) ϰ + A1
δ²P = 6 A3 ϰ + 2 A2
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| CALC : Math : Polynomial 014 : Fourth Degree Quartic Polynomial Single Real Root Or Zero : Calculator
Description : Calculate a single real root or zero of a quartic fourth order polynomial.
Discussion : The tool calculates the root (value equals zero), the maximum or minimum point (slope equals zero) and the inflection point (curvature equals zero). The root of the polynomial must be real. The guess value should be close to the root value, maximum minimum value, or inflection value. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and to estimate the guess values. The tool uses Newtons method to find the root. The functions PolyRoots(), PolyMaxMin() and PolyInflection calculate the polynomial roots, the polynomial maxima minima and the polynomial inflection points respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
- g = Initial Guess
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Root Or Zero
- I = Inflection Point
- M = Maxima Or Minima
- P = Output Value
Calculation :
C = PolyRoot( A , g )
M = PolyMaxMin( A , g )
I = PolyInflection( A , g )
P = ( ( ( A4 ϰ + A3 ) ϰ + A2 ) ϰ + A1 ) ϰ + A0
δP = ( ( 4 A4 ϰ + 3 A3 ) ϰ + 2 A2 ) ϰ + A1
δ²P = ( 12 A4 ϰ + 6 A3 ) ϰ + 2 A2
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| CALC : Math : Polynomial 015 : Fifth Degree Quintic Polynomial Single Real Root Or Zero : Calculator
Description : Calculate a single real root or zero of a quintic fifth order polynomial.
Discussion : The tool calculates the root (value equals zero), the maximum or minimum point (slope equals zero) and the inflection point (curvature equals zero). The root of the polynomial must be real. The guess value should be close to the root value, maximum minimum value or inflection value. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. Use the plot pages to plot the polynomial and to estimate the guess values. The tool uses Newtons method to find the root. The functions PolyRoots(), PolyMaxMin() and PolyInflection calculate the polynomial roots, the polynomial maxima minima and the polynomial inflection points respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
- g = Initial Guess
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Root Or Zero
- I = Inflection Point
- M = Maxima Or Minima
- P = Output Value
Calculation :
C = PolyRoot( A , g )
M = PolyMaxMin( A , g )
I = PolyInflection( A , g )
P = ( ( ( ( A5 ϰ + A4 ) ϰ + A3 ) ϰ + A2 ) ϰ + A1 ) ϰ + A0
δP = ( ( ( 5 A5 ϰ + 4 A4 ) ϰ + 3 A3 ) ϰ + 2 A2 ) ϰ + A1
δ²P = ( ( 20 A5 ϰ + 12 A4 ) ϰ + 6 A3 ) ϰ + 2 A2
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| CALC : Math : Polynomial 016 : Nth Degree Polynomial Single Real Root Or Zero : Calculator
Description : Calculate a single real root or zero of an nth order polynomial.
Discussion : Enter the polynomial coefficients as a comma separated list in order A0, A1, A2....An. The tool calculates the root (value equals zero), the maximum or minimum point (slope equals zero) and the inflection point (curvature equals zero). The root of the polynomial must be real. The tool also calculates the polynomial value, slope and curvature at an input value ϰ. The guess value should be close to the root value, maximum minimum value or inflection value. Use the plot pages to plot the polynomial and to estimate the guess values. The tool uses Newtons method to find the root. The functions PolyRoots(), PolyMaxMin() and PolyInflection calculate the polynomial roots, the polynomial maxima minima and the polynomial inflection points respectively.
Input Variables :
- ϰ = Input Value
- A = Equation Coefficients
- g = Initial Guess
Output Variables :
- δ²P = Curvature
- δP = Slope Or Derivative
- C = Root Or Zero
- I = Inflection Point
- M = Maxima Or Minima
- P = Output Value
Calculation :
P = PolyValue( A , ϰ )
δP = PolySlope( A , ϰ )
δ²P = PolyCurve( A , ϰ )
C = PolyRoot( A , g )
M = PolyMaxMin( A , g )
I = PolyInflection( A , g )
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