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# 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 |

vCalc.4th Degree Polynomial

The **4 ^{th} 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:

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