3rd Degree Polynomial

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vCalc.3rd Degree Polynomial
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The 3rd Degree Polynomial calculator computes a answer (y) to third degree polynomial where a, b, c, and  d are each multiplicative constants and x is the independent variable.

INSTRUCTIONS: Enter the following:

• 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

3rd Degree Polynomial (y):  The calculator returns the value of y.

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:

1.    a*x^3
2.    b*x^2
3.    c*x
4.   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, and can also be zero.  Technically, the 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 = 5x^3 + 0x^2 + 0x + 23 = 5x^3 + 23
•        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