What are the steps for solving a polynomial equation?

Answer 1

Get all the terms on one side of the equation, leaving the other side zero.

If the highest exponent is 2, the quadratic formula will suffice.

If the highest exponent is 3, the cubic formulas will suffice.

The quadratic and cubic formulas can be found at

http://www.math.vanderbilt.edu/~schectex/courses/cubic/

If the highest exponent is 4, the quartic formulas will suffice. These are found at http://planetmath.org/encyclopedia/QuarticFormula.html

(Good Luck!)

If the highest exponent is greater than 4, there is no procedure for solving the equation, BUT because all polynomials are continuous, it is possible to find a number closer to a solution than any number selected (if there is a real solution). Therefore, an approximation of real solutions is possible.

For example: 2X^7-3X^4+X^3-1.5X+14=0

let 2x^7-3x^4+x^3-1.5x+14=y

let x=0 ... y=14 (positive; now we need a negative output)

let x=-2 ... y=-295 (so a solution X lies between -2 & 0)

let x=-1 ... y=9.5 POSITIVE (-2

let x=-1.5 ... y=-36.48... NEGATIVE (-1.5

let x=-1.25 ... y=-2.939... NEGATIVE (-1.25

let x=-1.2 ... y=.684... POSITIVE (-1.25

This process is infinitely repeatable, allowing us to get closer and closer to a solution (note that with only the information above, there may be more than one solution in the found range).

This method is similar to graphing the function

y=2X^7-3X^4+X^3-1.5X+14 on a graphing calculator and zooming in to where the curve crosses the X-axis.