# Polynomial Equation

How to solve a polynomial equation by using the synthetic division: 1 example and its solution.

## Examplex^{4} + 4x^{3} - 3x^{2} - 10x + 8 = 0

Factor the left side

by using the synthetic division.

Factor Theorem

f(x) = x^{4} + 4x^{3} - 3x^{2} - 10x + 8

Write the coefficients of the terms

in descending order:

1 4 -3 -10 8.

Draw an L shape form like this.

Pick a number

that seems to make the remainder 0.

1 seems to be good.

So write 1

on the left side of the form.

↓: 1 = 1

↗: 1⋅1 = 1

↓: 4 + 1 = 5

↗: 5⋅1 = 5

↓: -3 + 5 = 2

↗: 2⋅1 = 2

↓: -10 + 2 = -8

↗: -8⋅1 = -8

↓: 8 - 8 = 0

The remainder is 0.

So 1 is the right number.

1 5 2 -8 means

x^{3} + 5x^{2} + 2x - 8.

This seems to be factorable.

So do the synthetic division again.

Draw an L shape form like this.

Pick a number

that seems to make the remainder 0.

1 seems to be good.

So write 1

on the left side of the form.

↓: 1 = 1

↗: 1⋅1 = 1

↓: 5 + 1 = 6

↗: 6⋅1 = 6

↓: 2 + 6 = 8

↗: 8⋅1 = 8

↓: -8 + 8 = 0

The remainder is 0.

So 1 is the right number.

1 6 8 means

x^{2} + 6x + 8.

This seems to be factorable.

So do the synthetic division again.

Draw an L shape form like this.

Pick a number

that seems to make the remainder 0.

-2 seems to be good.

So write -2

on the left side of the form.

↓: 1 = 1

↗: 1⋅(-2) = -2

↓: 6 - 2 = 4

↗: 4⋅(-2) = -8

↓: 8 - 8 = 0

The remainder is 0.

So -2 is the right number.

1 4 means

x + 4.

(x + 4) is a binomial factor.

So stop finding the factors.

The left side numbers are

1, 1, and -2.

So write the factors

(x - 1)^{2}(x + 2).

1 4 means

x + 4.

So write (x + 4).

Write = 0.

So the given equation is

(x - 1)^{2}(x + 2)(x + 4) = 0.

Solve the equation

by finding the zeros of each factor.

The zero of (x - 1)^{2} is 1.

The zero of (x + 2) is -2.

The zero of (x + 4) is -4.

So x = 1, -2, -4.

So x = 1, -2, -4.