Synthetic Division

Synthetic Division

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

Example 1

Simplify the given expression. (x^3 - 7x + 11)/(x - 2)

Write the coefficients of the polynomial
in descending order:
1, 0, -7, 11.

Don't forget to write 0x2 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 x2 + 2x - 3.
It's the quotient.

5 is the remainder.

So (given) = x2 + 2x - 3 + 5/(x - 2).

Check the same example
solved by using long division.
You can see that the answers are the same.

Example 2

Simplify the given expression. (2x^4 + x^3 - 5x^2 + 3x + 4)/(x + 1)

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 2x3 - x2 - 4x + 7.
It's the quotient.

-3 is the remainder.

So (given) = 2x3 - x2 - 4x + 7 - 3/(x + 1).