Synthetic Division Made Easy: Step-by-Step Method

Synthetic division is a streamlined technique for dividing a polynomial by a linear binomial of the form . It strips away the variable notation entirely, leaving you with a compact grid of numbers that delivers the quotient and remainder in a fraction of the time that traditional long division would take. If you regularly factor polynomials, evaluate them at specific points, or hunt for rational zeros, synthetic division is one of the most practical shortcuts you can learn.

What Is Synthetic Division?

In standard polynomial long division you write out the dividend and divisor much like you would with whole numbers, aligning terms by degree, subtracting at each step, and bringing down the next term. It works perfectly well, but it involves writing the variable and its powers repeatedly, which makes the process slow and error-prone for higher-degree polynomials.

Synthetic division achieves the same result using only the coefficients. The method was popularised in the 19th century as a quick mental or pencil-and-paper shortcut. It applies whenever the divisor is a monic linear binomial, that is, a binomial of the form where the leading coefficient on is 1. If the divisor were , you would first factor out the 2 and adjust afterwards (or fall back to long division).

When to Use Synthetic Division

• Dividing by a linear factor such as or (which is ).
• Evaluating a polynomial at a point via the Remainder Theorem (see below).
• Testing possible rational zeros of a polynomial quickly.
• Deflating a polynomial once a root is found, reducing the degree by one so you can continue finding other roots.

It is not suitable when the divisor is a higher-degree polynomial (e.g. ) or a non-monic linear binomial (e.g. ) unless you adapt the technique. For those cases, traditional long division or another method is more straightforward.

The Step-by-Step Method

Suppose you want to divide the polynomial by . Here is the procedure:

1. Write the value of on the left. If the divisor is , write 3. If the divisor is , write .
2. List the coefficients of the dividend in order from the highest degree to the constant term. If any degree is missing, insert 0 as a placeholder.
3. Bring down the leading coefficient to the bottom row.
4. Multiply the value you just brought down by and write the product under the next coefficient.
5. Add the column (the coefficient plus the product) and write the sum in the bottom row.
6. Repeat steps 4 and 5 until you have processed every coefficient.
7. The bottom row now contains the coefficients of the quotient (one degree lower than the dividend) followed by the remainder as the last entry.

Synthetic Division vs Long Division

Both methods produce the same quotient and remainder. The differences are practical:

• Speed: Synthetic division is significantly faster for linear divisors because you never write the variable or its powers.
• Compactness: The working fits on a single line or two, making it easier to check.
• Scope: Long division handles any divisor of any degree, while synthetic division is limited to monic linear divisors.
• Readability: Some learners find the long-division layout more intuitive because it mirrors whole-number division. Synthetic division can feel abstract until you have practised it a few times.

The Remainder Theorem

The Remainder Theorem states that when a polynomial is divided by , the remainder equals . In other words, you can evaluate the polynomial at simply by running synthetic division and reading off the last number in the bottom row.

This is often faster than substituting directly into the polynomial, especially for high-degree expressions where many powers must be computed.

The Factor Theorem

The Factor Theorem is a direct consequence of the Remainder Theorem. It says: is a factor of if and only if . So when synthetic division produces a remainder of 0, you have confirmed that is a root and divides the polynomial exactly.

Worked Example 1: Degree 3 Polynomial

Divide by .

Step 1. The value of is 2. The coefficients are 2, 3, , 7.

Step 2. Set up the synthetic division grid:

Working through it:

• Bring down 2.
• . Add to 3 to get 7.
• . Add to to get 9.
• . Add to 7 to get 25.

Result: The quotient is and the remainder is 25.

By the Remainder Theorem, . You can verify: . It checks out.

Worked Example 2: Finding a Zero of a Degree 4 Polynomial

Let . Test whether is a zero.

The coefficients are 1, , 11, , 0 (constant term is 0).

The remainder is 0, so is indeed a root. The deflated polynomial is , which you can factor further:

Therefore the full factorisation is:

The zeros are .

Worked Example 3: Missing Terms and a Negative Divisor

Divide by .

Since the divisor is , we use . The coefficients, including placeholders for the missing terms, are 3, 0, , 0, 8.

Working through it:

• Bring down 3.
• . Add to 0 to get .
• . Add to to get 46.
• . Add to 0 to get .
• . Add to 8 to get 744.

Result: The quotient is with a remainder of 744.

Notice how important the zero placeholders are. Without them the columns would be misaligned and the answer would be wrong.

Worked Example 4: Degree 4 with Exact Division

Divide by .

The remainder is 0, confirming that is a root. The quotient is . You could now apply synthetic division again with other suspected roots to continue factorising.

Testing on the quotient:

Remainder 0 again, so is also a root. The remaining quadratic is . The full factorisation is:

Finding Rational Zeros Systematically

The Rational Root Theorem tells you that any rational zero of a polynomial with integer coefficients must satisfy: divides the constant term and divides the leading coefficient. This gives you a finite list of candidates, each of which you can test quickly using synthetic division.

For example, consider . The constant term is 12 and the leading coefficient is 2.

• Divisors of 12:
• Divisors of 2:
• Possible rational zeros:

Testing via synthetic division:

Remainder 0, so is a root. The deflated polynomial factors as , giving roots and .

Common Mistakes to Avoid

• Forgetting zero placeholders. If the polynomial is , the coefficients must be written as 1, 0, 5, because the term has coefficient 0.
• Wrong sign for . When dividing by , use , not 3.
• Applying it to non-linear divisors. Synthetic division in its basic form only works for divisors of the form .
• Arithmetic errors. Because the method is so compact, a single addition or multiplication mistake propagates through every subsequent column. Double-check each step.

Frequently Asked Questions

Can synthetic division be used with complex numbers?

Yes. If is a complex number such as , the procedure is identical. You multiply and add complex numbers instead of real ones. This can be useful when finding complex roots, though in practice most students first encounter synthetic division with real numbers.

What happens if the leading coefficient of the divisor is not 1?

Standard synthetic division requires a monic divisor (leading coefficient 1). If you need to divide by , rewrite it as , perform synthetic division by , and then divide the resulting quotient coefficients by 2.

How is synthetic division related to Horner's method?

They are essentially the same algorithm. Horner's method rewrites a polynomial in nested form: . Evaluating this nested form at produces exactly the same sequence of multiplications and additions as synthetic division.

Can I use synthetic division for polynomial quotients with remainders?

Absolutely. The last number in the bottom row is the remainder. If it is non-zero, you express the result as . This is the standard division algorithm for polynomials.

Is synthetic division useful for finding all roots of a polynomial?

It is one of the most efficient tools for that purpose. Combined with the Rational Root Theorem, you generate a list of candidate rational zeros, test each one with synthetic division (a remainder of 0 confirms it is a root), and then deflate the polynomial to a lower degree. Once you reach a quadratic, you can finish with the quadratic formula.