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# 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 |

vCalc.3rd Degree Polynomial

The **3 ^{rd} 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:

- `a*x^3`
- `b*x^2`
- `c*x`
- `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**, **c ** and **d **can also be zero. Technically, the **a **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 = 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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**3rd Degree Polynomial**, is used in 2 calculators and 1 equation/constant.

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