Remainder Theorem Calculator
Evaluate a cubic at x = a to get the remainder instantly.
Enter your values
Enter the coefficient of x³ you are working with.
Enter the coefficient of x² you are working with.
Enter the coefficient of x you are working with.
Enter the constant term you are working with.
Enter the divide by (x − a), a = you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Remainder P(a)
- 0
- Is (x − a) a factor?
- Yes — remainder is zero
- Quotient
- 1x² + -1x + -2
- a used
- 3
Working
- 1Synthetic division brings down each coefficient, multiplies by a and adds to the next.
- 2Final value = P(3) = 0
Quick answer
The remainder theorem says dividing P(x) by (x − a) leaves a remainder of exactly P(a), so a remainder of zero means (x − a) is a factor.
P(x) ÷ (x − a) leaves remainder P(a)
At a glance
| What it does | Evaluate a cubic at x = a to get the remainder instantly. |
|---|---|
| Category | Math |
| Inputs needed | Coefficient of x³, Coefficient of x², Coefficient of x, Constant term, Divide by (x − a), a = |
| Main output | Remainder P(a) |
| Formula | P(x) ÷ (x − a) leaves remainder P(a) |
| Cost | Free — no sign-up, no download |
How to use the Remainder Theorem Calculator
- 1Enter the four coefficients of your cubic, using 0 for missing terms.
- 2Enter the value a from the divisor (x − a).
- 3A remainder of zero means you have found a root and the quotient factors further.
Inputs explained
- Coefficient of x³
- Enter the coefficient of x³ you are working with.
- Coefficient of x²
- Enter the coefficient of x² you are working with.
- Coefficient of x
- Enter the coefficient of x you are working with.
- Constant term
- Enter the constant term you are working with.
- Divide by (x − a), a =
- Enter the divide by (x − a), a = you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Coefficient of x³1
- Coefficient of x²-4
- Coefficient of x1
- Constant term6
- Divide by (x − a), a =3
Results
- Remainder P(a)0
- Is (x − a) a factor?Yes — remainder is zero
- Quotient1x² + -1x + -2
- a used3
- Synthetic division brings down each coefficient, multiplies by a and adds to the next.
- Final value = P(3) = 0
Frequently asked questions
What is the factor theorem?
The special case where the remainder is zero: (x − a) divides P(x) exactly, so a is a root.
Can I use a quadratic?
Yes — set the x³ coefficient to 0 and the quotient becomes linear.
What do I need to enter into the Remainder Theorem Calculator?
Just 5 values: coefficient of x³, coefficient of x², coefficient of x, constant term, divide by (x − a), a =. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Remainder Theorem Calculator work out the answer?
The remainder theorem says dividing P(x) by (x − a) leaves a remainder of exactly P(a), so a remainder of zero means (x − a) is a factor. It applies the formula P(x) ÷ (x − a) leaves remainder P(a) and shows the working so you can check each step by hand.
Is the Remainder Theorem Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.