How to Divide Polynomials?

Dividing polynomials is an operation in which a polynomial is divided by another polynomial, usually to a lesser degree as compared to the dividend. Dividing two polynomials may or may not lead to polynomials.

Related Topics

• The Remainder Theorem

A step-by-step guide to dividing polynomials

Polynomials are algebraic expressions made up of variables and coefficients. The phrase with the highest degree is placed first, followed by the lower terms.

Polynomial division is an algorithm for solving a rational number that represents a polynomial divided by a monomial or another polynomial. The divisor and the dividend are the same as we do for regular division.

Dividing polynomials by monomials

When dividing polynomials by monomials, the division can be done in two ways:

Splitting the terms method:

Split the terms of the polynomial separated by the operator \(+\) or \(-\) between them and simplify each sentence.

Factorization method:

When dividing polynomials, you may have to factor in polynomials to find a common factor between the numerator and the denominator.

Dividing polynomials using long division

Let’s review the algorithm for dividing polynomials by binomials using an example:

1. Divide the first term of the dividend \((4x^2)\) by the first term of the divisor \((x)\) and place it as the first term in the quotient \((4x)\).
2.  Multiply the divisor by that answer, and place the product \((4x^2- 12x)\) below the dividend.
3. Subtract to create a new polynomial \((7x-21)\).
4. Repeat the same process with the new polynomial after subtraction.

So, when divide a polynomial \((4x^2 – 5x – 21)\) with a binomial \((x – 3)\), the quotient is \(4x+7\) and the remainder is \(0\).

Dividing polynomials using synthetic division

Synthetic division is a technique for dividing a polynomial by a linear binomial by considering only the values of the coefficients. In this method, we first write polynomials in standard form, including the highest degree to the lowest degree.

When writing with descending powers, use \(0\) as the coefficient of the missing phrases. For example, \(x^3+3\) has to be written as \(x^3+ 0x^2+ 0x + 3\).

Follow these steps to divide a polynomial using the synthetic division method:

Let us divide \(x^2+ 3\) by \(x – 4\).

1. Write the divisor in the form of \(x – k\) and write \(k\) on the left side of the division. Here, the divisor is \(x-4\), so the value of \(k\) is \(4\).
2. Adjust the division by writing the dividend coefficients on the right and \(k \) on the left.
3. Now, bring down the coefficient of the highest degree term of the dividend as it is. Here, the leading coefficient is \(1\) (coefficient of \(x^2)\).
4. Multiply \(k\) by that leading coefficient and write the product below the second coefficient to the left side of the dividend. So, we get, \(4×1=4\) that we will write below \(0\).
5. Add the numbers written in the second column. So, we get, \(0+4=4\).
6. Repeat the same multiplication process \(k\) with the number obtained in step five and write the product in the next column to the right.
7. Finally, we write the final answer, which will be one degree less than the dividend. So, in our dividends, the highest degree is \(x^2\), so in the quotient, the highest degree term will be \(x\). Therefore, the answer is obtained \(x+4+\frac{19}{x-4}\).

Exercises for Dividing Polynomials

Divide.

1. \(\color{blue}{\left(2x^2-5x-3\right)\div \left(x-3\right)}\)
2. \(\color{blue}{\left(2x^{^6}−5x^{^5}−4x^{^4}+10x^{^3}+6x^{^2}−17x+5\right)\div \left(2x−5\right)}\)
3. \(\color{blue}{\:8x^6+4x^5−14x^4−5x^3+x^2−2x\div 2x+1}\)
4. \(\color{blue}{\frac{27m^6+9m^4−81m^2}{9m^2}}\)

1. \(\color{blue}{2x+1}\)
2. \(\color{blue}{x^5-2x^3+3x-1}\)
3. \(\color{blue}{4x^5-7x^3+x^2-1+\frac{1}{2x+1}}\)
4. \(\color{blue}{3m^4+m^2-9}\)

Recommended EffortlessMath Books

For a workbook that walks through polynomial division with both long division and synthetic division, the Algebra II for Beginners covers the full unit with worked examples. For pre-calc-level coverage that connects polynomial division to rational functions and partial fractions, see the Pre-Calculus for Beginners.

Algebra II for Beginners 2026 The Ultimate Step by Step Guide to Acing Algebra II

Download

$27.99 Original price was: $27.99.$17.99Current price is: $17.99.

Rated 5.00 out of 5 based on 1 customer rating

Satisfied 1 Students

Frequently Asked Questions

What is polynomial division?

Polynomial division is the process of dividing one polynomial by another, just like numerical long division. The result is a quotient polynomial and possibly a remainder polynomial whose degree is lower than the divisor’s. Two main methods exist: long division (works for any divisor) and synthetic division (works only for linear divisors).

How do you do polynomial long division?

Just like numerical long division: divide, multiply, subtract, bring down. Divide the leading term of the dividend by the leading term of the divisor to get the first quotient term. Multiply the divisor by that term, subtract from the dividend, bring down the next term, and repeat until the remainder has lower degree than the divisor.

What is synthetic division?

Synthetic division is a shortcut for dividing a polynomial by a linear binomial of the form \(x-c\). You write only the coefficients of the dividend, drop down the leading coefficient, multiply by \(c\), add, and repeat. The last number is the remainder; the others form the quotient. Faster than long division when it applies.

When can you use synthetic division?

Only when the divisor is linear in the form \(x-c\) (or you can rewrite the divisor that way). If the divisor is \(x+3\), use \(c=-3\). If the divisor is \(2x-6\), factor out the 2 first to get \(2(x-3)\), then divide by \(x-3\) and remember to divide the quotient by 2 at the end.

What is the Remainder Theorem?

If a polynomial \(f(x)\) is divided by \(x-c\), the remainder equals \(f(c)\). So instead of performing the division, you can just plug \(c\) into the polynomial. Example: dividing \(f(x)=x^3-2x+5\) by \(x-2\) gives remainder \(f(2)=8-4+5=9\) without doing any division.

What is the Factor Theorem?

If \(f(c)=0\), then \(x-c\) is a factor of \(f(x)\) — and vice versa. This is the special case of the Remainder Theorem where the remainder is zero. Example: \(f(x)=x^3-7x+6\) has \(f(1)=1-7+6=0\), so \(x-1\) is a factor of \(f(x)\).

What do you do with missing terms in polynomial division?

Fill in zeros for the missing degrees. If your dividend is \(x^4-3\), rewrite it as \(x^4+0x^3+0x^2+0x-3\) before dividing. Skipping placeholder zeros makes the column alignment fall apart and almost guarantees a wrong answer.

How do you check polynomial division?

Multiply your quotient by the divisor and add the remainder. The result should be the original dividend: \(f(x)=q(x)\cdot d(x)+r(x)\). If it matches, your division is correct. If not, go back and find the slip — usually a sign error or a missed placeholder.

What is the degree of the remainder in polynomial division?

The remainder’s degree is always strictly less than the divisor’s degree. If you divide by a quadratic, the remainder is at most linear. If you divide by a linear binomial, the remainder is a constant. Stop dividing as soon as the running remainder’s degree drops below the divisor’s.

Why do we divide polynomials?

To factor higher-degree polynomials, find rational roots, simplify rational expressions, and prepare polynomials for partial-fraction decomposition in calculus. The Factor Theorem turns root-finding into a division problem — and polynomial division is how you finish the factoring once you find one root.

Related EffortlessMath Lessons

If a topic on this page feels rusty, these short lessons go deeper:

• The Remainder Theorem
• How to use synthetic division
• Factoring polynomials
• Rules of exponents
• Order of operations (PEMDAS)