# Synthetic Division

How to solve dividing polynomial problems by using synthetic division: examples and their solutions.

## Example 1

Write the coefficients of the polynomial

in descending order:

1, 0, -7, 11.

Don't forget to write 0*x*^{2} term.

Ascending order, descending order

Draw an L shape form.

Write the zero of the divisor (2)

on the left of the form.

Starting from the top left 1,

add numbers in ↓ direction,

and multiply the divisor (2) in ↗ direction.

↓: 1 = 1↗: 1⋅2 = 2

↓: 0 + 2 = 2↗: 2⋅2 = 4

↓: -7 + 4 = -3↗: -3⋅2 = -6

↓: 11 - 6 = 5

1, 2, -3 means *x*^{2} + 2*x* - 3.

It's the quotient.

5 is the remainder.

So (given) = *x*^{2} + 2*x* - 3 + 5/(*x* - 2).

Check the same example

solved by using long division.

You can see that the answers are the same.

## Example 2

Write the coefficients of the polynomial

in descending order:

2, 1, -5, 3, 4.

Ascending order, descending order

Draw an L shape form.

Write the zero of the divisor (-1)

on the left of the form.

Starting from the top left 2,

add numbers in ↓ direction,

and multiply the divisor (-1) in ↗ direction.

↓: 2 = 2↗: 2⋅(-1) = -2

↓: 1 - 2 = -1↗: (-1)⋅(-1) = 1

↓: -5 + 1 = -4↗: -4⋅(-1) = 4

↓: 3 + 4 = 7↗: 7⋅(-1) = -7

↓: 4 - 7 = -3

2, -1, -4, 7 means 2*x*^{3} - *x*^{2} - 4*x* + 7.

It's the quotient.

-3 is the remainder.

So (given) = 2*x*^{3} - *x*^{2} - 4*x* + 7 - 3/(*x* + 1).