Solving Polynomial Equations

Solving Polynomial Equations

How to solve polynomial equations by using the synthetic division: example and its solution.

Example

If f(a) = 0, then (x - a) is the factor of f(x).

Factor the polynomial
to find the roots of the equation.

The roots are the numbers
that make the remainder of the synthetic division 0.

Use the synthetic division.
(divisor's zero: 1)

The remainder is 0.
So 1 is the root of the equation.

Factor theorem

Use the synthetic division.
(divisor's zero: 1)

The remainder is 0.
So another 1 is the root of the equation.

Use the synthetic division.
(divisor's zero: -2)

The remainder is 0.
So -2 is the root of the equation.

Use the synthetic division.
(divisor's zero: -4)

The remainder is 0.
So -4 is the root of the equation.

When doing the synthetic division
to solve polynomial equations,

it's good to use the division thorough
(until the constant term remains)

to prevent finding the wrong root:

you might say that 4 is the root, which is wrong.

The roots are 1, 1, -2, and 4.

So x = 1, -2, -4.