# 5th Degree Polynomial

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Equation / Last modified by KurtHeckman on 2017/02/15 18:14
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vCalc.5th Degree Polynomial
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The 5th Degree Polynomial equation computes a fifth degree polynomial where a, b, c, d, e ,and  f are each multiplicative constants and x is the independent variable.

INSTRUCTIONS: Enter the following:

• a - the multiplicative constant for the x^5 term
• b - the multiplicative constant for the x^4 term
• c - the multiplicative constant for the x^3 term
• d - the multiplicative constant for the x^2 term
• e - the multiplicative constant for the x term
• d - 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

The calculator returns the solution.  Note, you can enter values for the constants (a through d) and enter different values for x without re-entering the constants.

## DERIVATION

A fifth degree polynomial is an equation of the form:

y = ax^5 + bx^4 + cx^3 +dx^2 + ex + f    (showing the multiplications explicitly: y = a*x^5 + b*x^4 + c*x^3 +d*x^2 + e*x + f )

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

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

The equation is called "fifth degree" because it has an x^5 term.  If it also had an x^6 term in it, it would be called a sixth degree polynomial.

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

• where a is a constant multiplying the x^5 term
• where b is a constant multiplying the x^4 term
• where c is a constant multiplying the x^3 term
• where d is a constant multiplying the x^2 term
• where e is a constant multiplying the x^1 term (x to the first power is just x)
• where f is a constant multiplying the x^0 term (x to the zero power is just 1)

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

y = a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x^1 + f*x^0

## EXAMPLES

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

ex. y = 17x^5 + 5x^4 + 3x^3 + 6x^2 + 7x + 23  ; if x = 5, this evaluates to  y = 56833

ex. y = 1.8x^5 + 1.1x^4 + 3.7x^3 + 2.1x^2 + 0.4x + 22.2  ;  if x = 4.5, this evaluates to y = 4176.262500

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 "fifth 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^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f = 0*x^7 + 0*x^6 + a*x^5 + b*x^4 + c*x^3 + d*x^2 + e*x + f

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

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

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

•    ex. y = 2.5x^5 + 1.5x^4 - 2.5x^3 + 3.7x^2 + 1.4x + 22.2
•    ex. y = 2x^5 + 2x^4 - 7x^3 + 7x^2 + 4x + 2