# Solving Polynomial Equations

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

## Example: Solve x4 + 4x3 - 3x2 - 10x + 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.