Solving Polynomial Equations

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

a_n xⁿ + a_{n - 1} x^{n - 1}+ a_{n - 2} x^{n - 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: 

• (x - a) is a factor of f(x)

• a is a zero of the function f(x)

• 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 ± to indicate both the positive square root and the negative square root as solutions. So the solutions are x = ±.

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 È$ 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.