4th Degree Polynomial

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Equation / Last modified by KurtHeckman on 2015/07/09 15:55
`y = `
4th Degree Polynomial
Variable Instructions Datatype
`(a) x^4 " term coefficient"` Enter the multiplier of the fourth power term as any real number Decimal
`(b) x^3 " term coefficient"` Enter the multiplier of the third power term as any real number Decimal
`(c) x^2 " term coefficient"` Enter the multiplier of the squared term as any real number Decimal
`(d) x " term coefficient"` Enter the multiplier of the x term as any real number Decimal
`(e) "constant term"` Enter the additive constant as any real number Decimal
`(x) "Independent variable value"` Enter the value to be substituted for x in the 4th power term, the 3rd power term, the squared term and the linear x term Decimal
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vCalc.4th Degree Polynomial
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e6cc668e-da27-11e2-8e97-bc764e04d25f

The 4th Degree Polynomial equation computes a fourth degree polynomial where a, b, c, d, and  e are each multiplicative constants and x is the independent variable.  

INPUTS

This equation has the following inputs:

  • a - the multiplicative constant for the `x^4` term
  • b - the multiplicative constant for the `x^3` term
  • c - the multiplicative constant for the `x^2` term
  • d - the multiplicative constant for the `x` term
  • e - the constant term
  • x - the value to be substituted for x in the `x^4` term, `x^3` term,  the `x^2` term, and the x term

DERIVATION

A fourth degree polynomial is an equation of the form:

       `y = ax^4 + bx^3 +cx^2 + dx + e`    (showing the multiplications explicitly: `y = a*x^4 + b*x^3 + c*x^2 + d*x + e`)

In this simple algebraic form there are five additive terms shown on the right of the equation:

  1.    `a*x^4`
  2.    `b*x^3`
  3.    `c*x^2`
  4.    `d*x`
  5.   `e`

The equation is called "fourth degree" because it has an `x^4` term.  If it also had an `x^5` term in it, it would be called a fifth degree polynomial.

Note that the five terms could be thought of as each term having a multiplying constant: a, b, c, d, and e

  • where a is a constant multiplying the `x^4` term
  • where b is a constant multiplying the `x^3` term
  • where c is a constant multiplying the `x^2` term
  • where d is a constant multiplying the `x^1` term (x to the first power is just x)
  • where e is a constant multiplying the `x^0` term (x to the zero power is just 1)

So, the form of the fourth degree polynomial could be re-written more explicitly to be:

       `y = a*x^4 + b*x^3 + c*x^2 + d*x^1 + e*x^0`

EXAMPLES

We can make up examples values of a, b, c, d, and e to form an infinite number of variations on a fourth degree polynomial.  The multiplicative constants,  a, b, c, d, and e, can have integer or decimal values:

       ex. `y = 5x^4 + 3x^3 + 6x^2 + 7x + 23`  ; if `x = 5`, this evaluates to  `y = 3708`

       ex. `y = 1.1x^4 + 3.7x^3 + 2.1x^2 + 0.4x + 22.2`  ;  if `x = 4.5`, this evaluates to `y = 854.75625`

The multiplicative constants for  b, c, and can also be zero.  Technically, the constant can't be zero if the equation is to be called a "fourth degree polynomial" but this brings up an interesting point: a lower degree polynomial is only a higher degree polynomial with all the higher order terms having multiplicative constants equal to zero. `a*x^4 + b*x^3 + c*x^2 + d*x + e = 0*x^6 + 0*x^5 + a*x^4 + b*x^3 + c*x^2 + d*x + e`

Here are two examples of the b, c, d, or e constants being zero.

  •        ex. `y = 5x^4 + 8x^3 + 0x^2 + 0x + 23 = 5x^4 + 8x^3 + 23`
  •        ex. `y = 11x^4 + 2x^3 + 0x^2 + 6.5x + 0 = 11x^4 + 2x^3 + 6.5x`

Here is an example of one of the constants, the c constant, being a negative multiplier:

  •    ex. `y = 1.5x^4 + 2.5x^3 - 3.7x^2 + 1.4x + 22.2`
  •    ex. `y = 2x^4 + 7x^3 - 7x^2 + 4x + 2`

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