Math Calculators
Polynomial Long Division Calculator
Polynomial Long Division Calculator
Enter Polynomials
Use ^ for exponents. Example: x^2 + 3x – 4
Example: x + 2 or x – 3
Final Answer
Step-by-Step Solution
How to Use This Calculator
Input Format:
- Use ^ for exponents (x^2, x^3)
- Use + and – for addition/subtraction
- Use * for multiplication (optional)
- Spaces are optional
Examples:
- Dividend: x^3 + 2x^2 – 5x + 6
- Divisor: x + 1
- Dividend: 2x^4 – 3x^2 + 7
- Divisor: x – 2
What is Polynomial Long Division?
Polynomial long division is a method used to divide one polynomial (the dividend) by another (the divisor), similar to how you divide numbers.
Why it matters:
- Simplifies complex expressions.
- Helps in algebra, calculus, and solving rational functions.
- Essential for factoring and understanding polynomial behavior.
The general result looks like this:
Division Formula
Dividend
÷
Divisor
=
Quotient
+
Remainder
Divisor
Example:
Problem: 17 ÷ 5
17 ÷ 5 = 3 +
2
5
Explanation:
- Dividend = 17
- Divisor = 5
- Quotient = 3 (how many times 5 goes into 17)
- Remainder = 2 (what’s left over: 17 – 15 = 2)
Check: 3 + 2/5 = 3 + 0.4 = 3.4
And 17 ÷ 5 = 3.4 ✓
Understanding the Terms
- Dividend: The polynomial you are dividing. Example: x3+2×2−5x+6x^3 + 2x^2 – 5x + 6×3+2×2−5x+6
- Divisor: The polynomial you are dividing by. Example: x+1x + 1x+1
- Quotient: The result of division without remainder.
Remainder: What is left after complete division; its degree is less than the divisor.
Step-by-Step Process
Let’s divide x3 + 2x2 – 5x + 6 by x + 1
Step 1: Divide the first term of the dividend by the first term of the divisor
x3 ÷ x = x2
• This becomes the first term of the quotient: x2
Step 2: Multiply the divisor by this quotient term
x2 · (x + 1) = x3 + x2
• Multiply each term of the divisor by x2
Step 3: Subtract this result from the dividend
(x3 + 2x2 – 5x + 6) – (x3 + x2) = x2 – 5x + 6
• Subtraction eliminates the highest-degree term.
Step 4: Bring down the next term (if necessary)
• Already included: x2 – 5x + 6
• Highest-degree term now: x2
Step 5: Repeat the process
Divide x2 ÷ x = x → next quotient term: x
Multiply divisor:
x · (x + 1) = x2 + x
Subtract:
(x2 – 5x + 6) – (x2 + x) = -6x + 6
Step 6: Repeat again
Divide -6x ÷ x = -6 → next quotient term: -6
Multiply divisor:
-6 · (x + 1) = -6x – 6
Subtract:
(-6x + 6) – (-6x – 6) = 12
• Now remainder = 12, which has lower degree than divisor → stop.
Step 7: Write the final answer
x3 + 2x2 – 5x + 6 ÷ x + 1 = x2 + x – 6 +
12
x + 1
• Quotient = x2 + x – 6
• Remainder = 12
How the Calculator Helps
- Enter the dividend polynomial.
- Enter the divisor polynomial.
- Click Calculate → the calculator:
- Divides each term automatically.
- Shows multiplication and subtraction at each step.
- Outputs quotient + remainder clearly.
- Divides each term automatically.
You get step-by-step results without mistakes.
FAQs About Polynomial Long Division
Yes, the method works for any degree.
The dividend is exactly divisible; quotient is the final answer.
Synthetic division is quicker but works only for divisors like
x − c.
Yes, subtract carefully; signs matter.
Multiply the quotient by divisor and add remainder; it should equal the dividend.
Yes, especially with step-by-step guidance.
Insert them with zero coefficients to avoid mistakes.
Yes, it can have one or more terms but must have lower degree than divisor.
Yes, coefficients can be decimals or fractions.
It’s used in integration, partial fractions, and simplifying rational functions.
