3rd Degree Polynomial

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Equation / Last modified by Administrator on 2016/01/26 16:05
3rd Degree Polynomial
Variable Instructions Datatype
`(a)"Multiplier constant for "x^3" term"` Enter a real number to multiply the x-cubed term Decimal
`(b)"Multiplier constant for "x^2" term"` Enter a real number to multiply the x-squared term Decimal
`(c)"Multiplier constant for "x" term"` Enter a real number to multiply the x term Decimal
`(d)"Additive Constant term"` Enter a real number constant Decimal
`(x)"Independent variable"` Enter a real number to be substituted for x Decimal
Type
Equation
Category
Mathematics->Algebra
Contents
5 variables
Tags:
Rating
ID
vCalc.3rd Degree Polynomial
UUID
e6cc6337-da27-11e2-8e97-bc764e04d25f

The 3rd Degree Polynomial equation computes a third degree polynomial where a, b, c, and  d are each multiplicative constants and x is the independent variable.  

INPUTS

This equation has the following inputs:

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

DERIVATION

A third degree polynomial is an equation of the form:

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

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

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

The equation is called "third degree" because it has an `x^3` term.  If it also had an `x^4` term in it, it would be called a fourth degree polynomial.

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

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

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

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

EXAMPLES

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

       ex. `y = 5x^3 + 6x^2 + 7x + 23`  ; if `x = 5`, this evaluates to  `y = 833`

       ex. `y = 3.7x^3 + 2.1x^2 + 0.4x + 22.2`  ;  if `x = 4.5`, this evaluates to `y = 403.6875`

The multiplicative constants for  b, and can also be zero.  Technically, the constant can't be zero if the equation is to be called a "third 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^3 + b*x^2 + c*x + d = 0*x^5 + 0*x^4 + a*x^3 + b*x^2 + c*x + d`

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

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

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

  •    ex. `y = 2.5x^3 + 3.7x^2 - 1.4x + 22.2`
  •    ex. `y = 11x^3 + 7x^2 - 4x + 2`

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