2nd Degree Polynomial

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Equation / Last modified by Administrator on 2016/10/23 22:11
`"Result" = `
2nd Degree Polynomial
Variable Instructions Datatype
`(a)"Coefficient of " x^2` Enter a real number for the multiplier of the x-squared term Decimal
`(b)"Coefficient of " x` Enter a real number for the coefficient of the x term Decimal
`(c)"Constant Term"` Enter a real number of the constant term Decimal
`(x)"Value of independent x"` Enter a real number to be substituted for x Decimal
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vCalc.2nd Degree Polynomial
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e6cc5fd5-da27-11e2-8e97-bc764e04d25f

A second degree polynomial is where a, b, and c are each multiplicative constants and x is the independent variable with powers of zero (constant), one and two (squared).  

INPUTS

This equation has the following inputs:

  • a - the multiplicative constant for the `x^2` term
  • b - the multiplicative constant for the x term
  • c - constant term
  • x - the value to be substituted for x in both the the `x^2` term and the x term

DERIVATION

A second degree polynomial is an equation of the form:

       `y = ax^2 + bx + c`    (showing the multiplications explicitly: `y = a*x^2 + b*x + c`)

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

  1.    `a*x^2`
  2.    `b*x`
  3.   `c`

The equation is called "second degree" because it has an `x^2` term.  If it also had an `x^3` term in it, it would be called a third degree polynomial.

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

  • where a is a constant multiplying the `x^2` term
  • where b is a constant multiplying the `x^1` term (x to the first power is just x)
  • where c is a constant multiplying the `x^0` term (x to the zero power is just 1)

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

       `y = a*x^2 + b*x^1 + c*x^0`

EXAMPLES

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

       ex. `y = 5x^2 + 7x + 23`  ;  and if `x = 3`, `y = 89`

       ex. `y = 3.7x^2 + 0.4x + 22.2` ; and if `x = 6.5`, `y = 181.125`

The multiplicative constants for b and can also be zero.  Technically, the constant can't be zero if the equation is to be called a "second 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^2 + b*x + c = 0*x^5 + 0*x^4 + 0*x^3 + a*x^2 + b*x + c`

Here are two examples of the b and the c constants being zero.

  •       ex. `y = 5x^2 + 0x + 23 = 5x^2 + 23`
  •       ex. `y = 2x^2 + 6.5x + 0 = 2x^2 + 6.5x`

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

  •      ex. `y = 3.7x^2 - 1.4x + 22.2`
  •      ex. `y = 7x^2 - 4x + 2`