# Synthetic Division

How to divide a polynomial by a binomial by doing the synthetic division: 2 examples and their solutions.

## Example 1

### Example

### Solution

The synthetic division is a way

to divide a polynomial by a binomial,

just like the long division.

The numerator is

x^{3} + 0x^{2} - 7x + 11.

(If there's a missing term, like the x^{2} term,

think its coefficient as 0.)

Then write the coefficients of the numerator terms

in descending order:

1 0 -7 11.

Draw an L shape form like this.

On the left side,

write the zero of the denominator (x - 2), 2.

Then, do the synthetic division.

Write 1

in ↓ direction.

↓: 1 = 1

Multiply 1 and the left 2.

Write 1⋅2 = 2

in ↗ direction.

↗: 1⋅2 = 2

Add 0 and 2

in ↓ direction.

↓: 0 + 2 = 2

Multiply 2 and the left 2.

Write 2⋅2 = 4

in ↗ direction.

↗: 2⋅2 = 4

Add -7 and 4

in ↓ direction.

↓: -7 + 4 = -3

Multiply -3 and the left 2.

Write -3⋅2 = -6

in ↗ direction.

↗: -3⋅2 = -6

Add 11 and -6

in ↓ direction.

↓: 11 - 6 = 5

Draw another L shape form

that covers the right end number 5.

This 5 is the remainder.

This is the end of the calculation.

Let's write the answer from this.

The numbers below the L shape,

1 2 3,

are the coefficients of the quotient terms.

So the quotient is

x^{2} + 2x + 3.

The number in this L shape, 5,

is the remainder.

So write +, a fraction bar,

and 5 in the numerator.

The zero of the left 2 is

(x - 2).

So write (x - 2)

in the denominator.

From the synthetic division,

you can find that

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

So

x^{2} + 2x + 3 + 5/(x - 2)

is the answer.

## Example 2

### Example

### Solution

The numerator is

2x^{4} + x^{3} - 5x^{2} + 3x + 4.

Then write the coefficients of the numerator terms

in descending order:

2 1 -5 3 4.

Draw an L shape form like this.

On the left side,

write the zero of (x + 1), -1.

Then, do the synthetic division.

Write 2

in ↓ direction.

↓: 2 = 2

Multiply 2 and the left -1.

Write 2⋅(-1) = -2

in ↗ direction.

↗: 2⋅(-1) = -2

Add 1 and -2

in ↓ direction.

↓: 1 - 2 = -1

Multiply -1 and the left -1.

Write (-1)⋅(-1) = 1

in ↗ direction.

↗: (-1)⋅(-1) = 1

Add -5 and 1

in ↓ direction.

↓: -5 + 1 = -4

Multiply -4 and the left -1.

Write (-4)⋅(-1) = 4

in ↗ direction.

↗: (-4)⋅(-1) = 4

Add 3 and 4

in ↓ direction.

↓: 3 + 4 = 7

Multiply 7 and the left -1.

Write 7⋅(-1) = 4

in ↗ direction.

↗: 7⋅(-1) = -7

Add 4 and -7

in ↓ direction.

↓: 4 - 7 = -3

Draw another L shape form

that covers the right end number -3.

This -3 is the remainder.

This is the end of the calculation.

Let's write the answer from this.

The numbers below the L shape,

2 -1 -4 7,

are the coefficients of the quotient terms.

So the quotient is

2x^{3} - x^{2} - 4x + 7.

The number in this L shape, -3,

is the remainder.

So write -, a fraction bar,

and 3 in the numerator.

The zero of the left -1 is

(x + 1).

So write (x + 1)

in the denominator.

From the synthetic division,

you can find that

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

So

2x^{3} - x^{2} - 4x + 7 - 3/(x + 1)

is the answer.