# Solving Polynomial Equations

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

## Example: Solve *x*^{4} + 4*x*^{3} - 3*x*^{2} - 10*x* + 8 = 0

Factor the left side

by using synthetic division.

Factor theorem

Synthetic division

Write the coefficients of the terms:

1 4 -3 -10 8.

Write the L shape form like this.

Pick a number

that seems to make the remainder 0.

1 seems to be the zero.

So write 1

next to the form.

↓: 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 zero of the polynomial.

Do synthetic division again.

Pick a number

that seems to make the remainder 0.

1 seems to be the zero.

So write 1

next to the form.

↓: 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 zero of the polynomial.

Do synthetic division again.

Pick a number

that seems to make the remainder 0.

-2 seems to be the zero.

So write -2

next to the form.

↓: 1

↗: 1⋅(-2) = -2

↓: 6 - 2 = 4

↗: 4⋅(-2) = -8

↓: 8 - 8 = 0

The remainder is 0.

So -2 is the zero of the polynomial.

Do synthetic division again.

Pick a number

that seems to make the remainder 0.

-4 seems to be the zero.

So write -4

next to the form.

↓: 1

↗: 1⋅(-4) = -4

↓: 4 - 4 = 0

The remainder is 0.

So -4 is the zero of the polynomial.

1, 1, -2, and -4 are the zeros.

And the quotient is 1.

So (*x* - 1)^{2}(*x* - (-2))(*x* - (-4))⋅1 = 0.

The zeros of the factors are

1, -2 and -4.

So *x* = 1, -2, -4.