Remainder Theorem Calculator

Use this remainder theorem calculator to find the remainder when a polynomial is divided by a linear divisor such as x – c.

Algebra Polynomial remainder with steps

Polynomial coefficients Enter coefficients from highest degree to constant term. Example: 2, -3, 0, 5 means 2x^3 – 3x^2 + 0x + 5.

c value in x – c For x – 2, enter 2. For x + 3, enter -3.

Rounding Used for decimal or fraction inputs.

Remainder: 7

P(2) = 7, so the remainder when the polynomial is divided by x – 2 is 7.

What the Result Means

The remainder theorem says that if a polynomial P(x) is divided by x – c, the remainder is P(c). This calculator substitutes your c value into the polynomial and also shows the Horner method steps used to get the result.

Synthetic evaluation steps will appear here after calculation.

Remainder Theorem Formula

If P(x) is divided by x – c, then:

Remainder = P(c)

For a polynomial such as ax^3 + bx^2 + cx + d, substitute the divisor value into x. If the divisor is x – 2, use x = 2. If the divisor is x + 3, rewrite it as x – (-3), so use x = -3.

How to Use This Calculator

Write the polynomial coefficients in order from the highest power down to the constant term.
Use zero for any missing power. For example, x^3 + 4 has coefficients 1, 0, 0, 4.
Enter the c value from the divisor x – c.
Click Calculate Remainder to evaluate P(c).
Read the remainder and the step-by-step synthetic evaluation below the tool.

Worked Example

Find the remainder when 2x^3 – 3x^2 + 5 is divided by x – 2.

P(2) = 2(2^3) – 3(2^2) + 0(2) + 5 = 16 – 12 + 0 + 5 = 9

The remainder is 9. If your polynomial has a missing x term, you still enter its coefficient as 0 so the powers stay in the correct order.

Common Use Cases

Checking the remainder of polynomial division without doing full long division.
Testing whether x – c is a factor of a polynomial. If P(c) = 0, then x – c is a factor.
Solving algebra homework problems that ask for the remainder only.
Comparing synthetic substitution with polynomial long division.

Common Mistakes

Entering coefficients in the wrong order. They must go from highest degree to constant term.
Skipping missing powers instead of entering 0.
Using the wrong sign for the divisor. For x + 4, the c value is -4.
Confusing the quotient with the remainder. This calculator focuses on the remainder value P(c).

Limitations

This tool handles real-number coefficients, decimals, and simple fractions such as 1/2. It is designed for division by a linear divisor in the form x – c. It does not perform full polynomial division by quadratic or higher-degree divisors.

Related Calculators

For nearby algebra work, you may also find the Quadratic Equation Calculator, Scientific Calculator, and Fraction Calculator useful.

FAQ

What is the remainder theorem?

The remainder theorem states that the remainder when P(x) is divided by x – c is equal to P(c).

How do I enter x + 3 as the divisor?

Rewrite x + 3 as x – (-3), then enter -3 as the c value.

Can this calculator show if something is a factor?

Yes. If the remainder is 0, then x – c is a factor of the polynomial.

Why do I need to enter zeros for missing terms?

Zeros keep each coefficient aligned with the correct power of x. Without them, the calculator may read the polynomial as a different expression.

Does the calculator support fractions?

Yes, you can enter simple fractions such as 3/4 or -5/2 in the coefficient list or c value field.

Remainder Theorem Calculator