Synthetic Division

Synthetic division is a shortcut for dividing a polynomial by a linear divisor of the form $x-c$. You use the coefficients instead of writing full polynomial long division, so it is faster when the divisor has exactly that form.

The condition matters. Standard synthetic division works for divisors like $x-2$ or $x+5$, because those are really $x-c$. If the divisor is not in that form, do not force the shortcut.

When Synthetic Division Works

Use synthetic division when the divisor is linear and can be written as $x-c$. That includes $x-2$, $x+3$, and $x-\frac{1}{2}$.

If the divisor is $2x-3$, the usual setup does not apply directly. In that case, use polynomial long division or rewrite the problem with care.

What The Numbers Mean

Suppose you divide a polynomial $P(x)$ by $x-c$. Synthetic division turns that problem into repeated multiply-and-add steps using the number $c$ and the coefficients of $P(x)$.

The last number is the remainder. Every number before it is a coefficient of the quotient. By the Remainder Theorem, that last number is also $P(c)$.

How To Do Synthetic Division

To set it up:

1. Write the number $c$ on the left.
2. List the coefficients of the polynomial in descending powers of $x$.
3. Include a zero for any missing power.
4. Bring down the first coefficient.
5. Multiply by $c$, write the result under the next coefficient, and add.
6. Repeat until you reach the end.

If a term is missing, the zero is not optional. It keeps every power lined up correctly.

Synthetic Division Example

Divide $2x^3 - 3x^2 + 4x - 5$ by $x-2$.

Because the divisor is $x-2$, use $c=2$. The coefficients are $2$, $-3$, $4$, and $-5$.

Now do the multiply-and-add steps:

1. Bring down the first coefficient: $2$.
2. Multiply $2$ by $2$ to get $4$, then add to $-3$ to get $1$.
3. Multiply $1$ by $2$ to get $2$, then add to $4$ to get $6$.
4. Multiply $6$ by $2$ to get $12$, then add to $-5$ to get $7$.

So the bottom row is $2$, $1$, $6$, $7$. The quotient coefficients are $2$, $1$, and $6$, and the remainder is $7$.

That gives the quotient

$$2x^2 + x + 6$$

with remainder $7$. So

$$2x^3 - 3x^2 + 4x - 5 = (x-2)(2x^2 + x + 6) + 7$$

and the division result can also be written as

$$\frac{2x^3 - 3x^2 + 4x - 5}{x-2} = 2x^2 + x + 6 + \frac{7}{x-2}$$

Why Students Use Synthetic Division

Synthetic division is useful when you want a quicker way to divide by $x-c$, test a possible factor, or find a remainder fast.

It often appears with the Remainder Theorem and Factor Theorem. If the remainder is $0$, then $x-c$ is a factor of the polynomial.

Common Synthetic Division Mistakes

Using The Wrong Sign

For a divisor $x-2$, use $2$. For a divisor $x+2$, use $-2$. The sign changes because $x+2 = x-(-2)$.

Forgetting Missing Terms

If you divide $x^3 + 5x - 1$, the coefficient list is not $1, 5, -1$. It is

$$1,\ 0,\ 5,\ -1$$

because the $x^2$ term is missing.

Using The Shortcut On The Wrong Divisor

The basic setup is for $x-c$. If the divisor is something like $2x-3$, do not put $3$ on the left and continue as usual. Use another method unless you know the modified setup.

Forgetting What The Last Number Means

The final number is the remainder, not another quotient coefficient.

When You Will See It

Synthetic division shows up when you:

1. divide a polynomial by $x-c$
2. check whether a linear expression is a factor
3. need the remainder $P(c)$
4. want a faster alternative to long division for this special case

Try A Similar Problem

Divide $x^3 - 6x^2 + 11x - 6$ by $x-1$ using synthetic division. Then use the remainder to decide whether $x-1$ is a factor. If you want one more case, try the same polynomial with $x-2$ and compare the remainders.