Advanced Polynomial Remainder Theorem Calculator

Calculator Inputs

Coefficients in descending powers

Use commas, spaces, or semicolons. Example gives 2x⁴ - 3x³ + 5x - 7.

Evaluation point a

The divisor is interpreted as x - a.

Decimal precision

Controls result formatting and displayed steps.

Graph minimum x

Graph maximum x

Plot points

Higher values create smoother curves.

Reset

Formula Used

The polynomial remainder theorem says that dividing a polynomial P(x) by x - a gives a remainder equal to P(a). This means you only need substitution, not long division, to get the remainder.

For coefficients c₀, c₁, ..., c_n, Horner evaluation is:

b₀ = c₀
b_k = c_k + a · b_k-1 for k = 1 to n

The final value b_n is the remainder. All earlier b values form the quotient coefficients when dividing by x - a.

How to Use This Calculator

Enter the polynomial coefficients from highest power to constant term.
Type the value of a for the divisor x - a.
Choose your preferred decimal precision.
Set the graph range and number of plot points.
Press calculate to see the remainder, quotient, factor check, Horner table, and graph.
Use CSV or PDF export buttons to save the result summary and steps.

Example Data Table

Polynomial	Divisor	a	Remainder	Quotient	Factor?
2x³ - 3x² + 5	x - 2	2	9	2x² + x + 2	No
x⁴ - 6x² + 8x - 3	x - 1	1	0	x³ + x² - 5x + 3	Yes
3x² + 4x - 1	x + 2	-2	3	3x - 2	No
x⁵ - 1	x - 1	1	0	x⁴ + x³ + x² + x + 1	Yes

FAQs

1. What does the remainder theorem actually prove?

It proves that the remainder from dividing P(x) by x - a equals P(a). A single substitution gives the remainder instantly and avoids long polynomial division.

2. Why are coefficients entered in descending powers?

Horner evaluation and synthetic division depend on the order of coefficients. Descending powers keep every multiplication and carry step aligned correctly.

3. What happens if the remainder is zero?

A zero remainder means x - a is a factor of the polynomial. The calculator also labels this in the factor theorem result box.

4. Can this calculator handle missing terms?

Yes. Enter zero for any missing power. For example, x³ + 5x - 7 should be entered as 1, 0, 5, -7.

5. Does the theorem work for divisors like x + 3?

Yes. Rewrite x + 3 as x - (-3). Then use a = -3. The calculator handles this by accepting negative evaluation points.

6. Is the quotient shown after division useful?

Yes. The quotient helps verify the division result and supports factorization, root checking, and follow-up algebra work.

7. Why is a graph included in this calculator?

The graph shows the polynomial shape across your chosen interval and highlights the exact evaluated point, making the remainder calculation easier to interpret visually.

8. Can I use this for non-linear divisors?

No. The remainder theorem applies directly to divisors of the form x - a only. Quadratic or higher divisors need polynomial division or other methods.