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# 5th Degree Polynomial

vCalc.5th Degree Polynomial

The **5 ^{th} 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:

- `a*x^5`
- `b*x^4`
- `c*x^3`
- `d*x^2`
- `e*x`
- `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**, **d **and **e **can also be zero. Technically, the **a **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 = 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`

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