4th Degree Polynomial

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Equation / Last modified by KurtHeckman on 2015/07/09 15:55
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vCalc.4th Degree Polynomial
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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