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# Solving Polynomial Equations

A polynomial equation is an equation that can be written in the form:

an xn + an - 1 xn - 1+ an - 2 xn - 2 + ... + a1 x + a0 = 0.

We will learn to solve this type of equation both algebraically and graphically.

## Solving Polynomial Equations by Factoring

With this method, we use the zero product property, just as we did to solve quadratic equations. In general, if a number is a solution to a polynomial equation f(x) = 0, then the following are true:

â€¢ (x - a) is a factor of f(x)

â€¢ a is a zero of the function f(x)

â€¢ a is an x-intercept of the graph of f(x)

## Solving Polynomial Equations by the Root Method

Recall that to solve x2 = C, C 0, we take the square root of both sides and use the symbol Â± to indicate both the positive square root and the negative square root as solutions. So the solutions are x = Â± .

We can solve the equation x3 = C, where C is any real number, by taking the cube root of both sides. The solution to this equation would then be . There is only one real cube root of a number, so there is oe Ãˆ\$ only one solution to this equation.

In general, when n is even, there will be two real nth roots and when n is odd, there will be only one nth root. So the Root Method is:

The real solutions of the equation xn = C are found by taking the nth root of both sides: if n is odd, and if n is even

Many real world models involve polynomial equations which either are not factorable or are extremely difficult to factor. In these cases, we need to rely on Graphical Methods.