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
x3 + 0x2 - 7x + 11.
(If there's a missing term, like the x2 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
x2 + 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) = x2 + 2x + 3 + 5/(x - 2).

So
x2 + 2x + 3 + 5/(x - 2)
is the answer.

Example 2

Example

Solution

The numerator is
2x4 + x3 - 5x2 + 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
2x3 - x2 - 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) = 2x3 - x2 - 4x + 7 - 3/(x + 1).

So
2x3 - x2 - 4x + 7 - 3/(x + 1)
is the answer.