Polynomial Remainder Theorem | Formula

When a polynomial is divided by a linear polynomial, the remainder theorem is used to get the remaining. The remaining is the number of items left over after a particular number of items is divided into groups with an equal number of items in each group. After division, it is something that “remains.” Let’s have a look at the remainder theorem.

The remaining theorem goes like this: r = a obtain the remainder when a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k. (k). The remainder theorem allows us to calculate the remainder of any polynomial divided by a linear polynomial without actually performing the division algorithm steps.

Remainder Theorem

Everyone enjoys finding a shortcut, whether it’s for driving directions or for any other long chore. Finding a faster and more effective way to go to the same destination makes you happy because you’ve probably saved time, effort, and/or money. The remainder theorem is one of the more useful of these types of shortcuts in mathematics.

The Chinese remainder theorem is a theorem that proves that simultaneous linear congruences with coprime moduli have a unique solution. In its most basic form, the Chinese remainder theorem determines a number p that leaves provided remainders when divided by some given divisors.

What is the Remainder theorem?

The polynomial remainder theorem, also known as tiny Bézout’s theorem, is an application of Euclidean polynomial division in algebra. It asserts that the residue of a polynomial f(x) divided by a linear polynomial x – r equals f. (r). Specifically, the factor theorem states that x – r is a divisor of f(x) if and only if f(r)= 0.

Remainder theorem Calculator

If you’re seeking a free online tool to help you with math computations and polynomial expressions, look no further. You’ve come to the correct location. To make the remainder theorem concept easier to grasp, we’ve broken it down into steps. You’ll learn things like what the remainder theorem is and how to solve it.

The free Remainder Theorem Calculator utility quickly calculates the remainder of a complex polynomial expression. Simply enter the input divided polynomial and divisor polynomial into the appropriate input fields and press the calculate button to quickly examine the result.

Remainder theorem Definition

We can define the remaining theorem as follows: A theorem in algebra states that if f(x) is a polynomial in x, the remainder after dividing f(x) by x an is f. (a).

Remainder theorem Formula

The remainder theorem is written as p(x) = (x-c) × q(x) + r in its general form (x). To show the remainder theorem formula, we’ll use polynomials.

Reminder = p when p(x) is divided by (x-a) (a).


When p(x) is divided by (ax+b)

Reminder = p(-b/a).

Proof of Remainder Theorem:

Dividend = (Divisor Quotient) + Remainder, as you know.

p(x) = (x-c)q(x) + r = (x-c)q(x) + r = (x-c)q(x) + r = (x-c)q(x) + r = (x-c)q(x) +

Let’s use x=c as an example.

p(c) = (c-c)·q(c) + r

Or, p(c) = (0)·q(c) + r

p(c) = r

Hence, proved.

Remainder theorem number system

It is preferable to comprehend the concepts of Divisor, Dividend, Quotient, and Remainder before moving on to the remainder theorem of numbers.

Remainder Theorem

Remainder Theorem Rule – 1 (Fundamental)

Positive and negative reminders might be used to communicate the remainder of the phrase. Both positive and negative remainders are technically correct.

However, remainders are always non-negative by definition. As a result, the final solution should only be expressed as a positive number. Take the negative remaining to simplify the solution, and then put it into the positive remainder.


  • If you divide the negative remainder by a divisor, you’ll get a positive remainder.
  • When the positive residual is subtracted from the divisor, the result is a negative remainder.

Example – 1:  Find the remainder of the expression of 49/9

In this question,

Positive remainder =  +4

Negative remainder  = -5

As per the remainder theorem, the final answer is 4.

Example – 2:  Find the remainder of the expression of 107 / 9

In this question

Positive remainder =  +8

Negative remainder  = -1

Finding the negative remaining in this expression is much easier than finding the positive remainder.

So, if you take the negative remainder “-1” and multiply it by the divisor “9,” you get the final answer: – 1 + 9 = 8.

Remainder Theorem Rule – 2 (For long expressions)

The remainder of the formula [A×B + C×D]/N is the same as that of [AR×BR + CR×DR]/N.


AR is the remainder when “N” divides “A”.

BR is the remainder when “N” divides “B”.

CR is the remainder when “N” divides “C”.

DR is the remainder when “N” divides “D”.

Example:  Find the remainder  when 47 x 52 is divided by 6


Here remainder for 47/6  is -1 or 5  and  52/6  is -2 or 4

Case 1: -1 x -2 = 2/ 6, remainder = 2

Then, Case 2: -1 x 4  = -4 / 6, remainder = – 4 + 6 (add divisor) = 2

Then, Case 3: 5 x -2  = -10 / 6, right arrow  remainder = – 4 + 6 (add divisor) = 2

Case 4: 5 x 4  = 20 / 6, remainder =  2

As per the reduction calculation, we can choose the option.

Remainder Theorem Rule – 3 (Cancellation rule)

You should strive to cancel out as much of the numerator and denominator as possible to simplify the expression of the sum, with a final reminder to multiply the cancelled number to achieve the right residual.


Find the remainder of 54 divided by 4.

= 54/4 = 27 / 2  ( cancel out by 2 of the numerator and denominator  )

The remainder is of the final expression is 1

The final reminder for the given expression is = 1 x 2 = 2

Remainder Theorem Rule – 4 (remainder of a number with power )

There are two effective rules for dealing with big powers.

Case 1: The remainder will be 1 if we can formulate the expression in the form {(qx + 1)^n} / q.

The residual is 1 in this situation, regardless of how large the power “n” is.

Example:  Find the remainder when 8 divides (321)5687.


We can express 321 as [(8×40) + 1]

So, the remainder of the above question is (1)5687  = 1

Case 2: If we can express the expression in the form {(qx \ – 1)^n}/{q}, the remainder will become ( – 1 )n

If n is an even number, the residual will be 1, whereas if n is an odd number, the remainder will be 0. (q-1).


Find the remainder when 7 divides (146)56.


We can express 146 as [(7×21) – 1]


The remainder of the above question is (- 1)56  = 1


Find the remainder when 6 divides (269)57587


We can express 269 as [( 6x 45) – 1].


The remainder of the above question is (- 1)57587  =  – 1 = -1 + 6 = 5

Remainder theorem questions and answers

Example 1:

Find the remainder when 25 divides 73 x 75 x 78 x 57 x 194.

Solution: = {73 × 75 × 57 × 194} /{25}

= { -2 × 0 × 7 × 7 } /{25}

= 0

Remainder theorem shortcut tricks:

The remainder of the complete expression is Zero if the denominator is perfectly divided by any one number of given numerator expression values.

Example 2:

Find the remainder when 32 divides 84 + 98+ 197 + 240 + 140.


= { 84 + 98 + 197 + 240 +140 }/{ 32 }

= { -12 + 2 +5 +16 +12 }/{32}

= 759 / 32.

So the remainder of the above expression is 23.

Example 3:

Find the remainder when 17 divides 1753 X 1749 X 83 X 171.

Solution: = {1753 × 1749 × 83 × 171 }/{ 17 }

= { +2 × -2 × -2 × 1 }/{ 17 }

=  8/17.

 So the remainder of the above expression is 8.

Example 4:

Find the remainder when 6 divides 385.


Here we can write 6 as 3 x 2 so ( cancel out by 3 of the numerator and denominator )

= { (3) ^85 }/{ 3 × 2 }

Or, we can write it as { (3) ^(85 -1) }/{ 2 }

= { (2 +1) ^84}/{ 2 }

= ⅓.

The remainder is 1.

The final remainder of the given question = 1 x 3 = 3.

Example 5: Find the remainder when 96 divides 2^70.

Solution:  Here we can write 96 as 2^5 x 3 so ( cancel out by 2^5 of the numerator and denominator ).

= {2 ^70 }/{ 96 }

= { 2 ^70 }/ { 2^5 × 3 }

Or, we can write it as { 2 ^(70 – 5) }/{ 3 }

= { 2 ^65}/{ 3 }

= {(3-1) ^65}/{ 3 }

= -⅓.

So, The remainder is -1 + 3 = 2.

The final remainder of the above question = 2 x  25 = 2 x 32.

Remainder theorem application

By calculating the remainder, R, we can utilize the polynomial remainder theorem to assess f(r). Synthetic division is computationally easier than polynomial long division. It is more difficult than evaluating the function itself. As a result, we can use synthetic division and the polynomial remainder theorem to evaluate the function more “cheaply.”

Another application of the remainder theory is the factor theorem, which states that if the remainder is zero, then the linear divisor is a factor. We can apply the factor theorem repeatedly to factorise the polynomial.

How to Solve the Remainder of Polynomial Expressions Using the Remainder Theorem?

When a polynomial f(x) is divided by a linear factor in the form of x-a, the Remainder Theorem is applied. Follow the methods below and use them to solve the remainder of a polynomial expression in a matter of seconds.

Assume that the polynomial f(x) is the dividend and that the linear expression is the divisor.

The linear expression must be written as x-a.

The remaining polynomial will then become f. (a).

So, in the polynomial expression, substitute the c value and evaluate to get the remaining value.

The theorem of the Remainder

When a polynomial function f(x) is divided by a linear function x-c, the polynomial function’s remainder is always equal to f. (c).

We know that Dividend = (Divisor x Quotient ) + Remainder

Then, f(x) = (x-c) × q(x) + r(x)

Where f(x) is the polynomial function

x-c is the linear factor

q(x) quotient

r(x) is remainder.


Question: Solve (x^4 + 7x^3 + 5x^2 – 4x + 15) % (x + 2) using remainder theorem?


Given values are

f(x) = x^4 + 7x^3 + 5x^2 – 4x + 15

x + 2 is in the form of x – (-2).

Then c = -2

f(-2) = (-2)^4 + 7(-2)^3 + 5(-2)^2 – 4(-2) + 15

= 16 – 56 + 20 + 8 + 15

= 3.

The remainder of the given polynomial is 3.

How Does Remainder Theorem Work?

Let’s look at a general scenario to see how the remainder theorem works. Let a(x) be the dividend polynomial and b(x) be the linear divisor polynomial, and q(x) and r be the quotient and constant remainder, respectively. As a result, a(x) = b(x) q(x) + r.

Let’s use k to represent the zero of the linear polynomial b(x). As a result, b(k) = 0. We get a(k) = b(k) q(k) + r if we use x as k in the highlighted relation above.

Read Also: Alternate Interior Angles: Examples, Definition, Theorem

It’s worth noting that this is permissible because the starred relation remains true for all x values. It’s a polynomial identity, in reality. We’re left with a(k)=r because b(k)=0. In other words, when x equals k, the remainder equals the value of a(x). That’s exactly what we found! The remainder theorem is exactly what it sounds like: The remainder is obtained by r=a when a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x equal to k. (k). Consider the following example with two polynomials to demonstrate how it works in the case of polynomials:

a(x): 6×4 – x3 + 2×2 – 7x + 2

b(x): 2x + 3

On dividing polynomials, the quotient polynomial and the remainder are:

q(x) = 3×3 – 5×2 + 17/2 x – 65/4 r = 203/4

We calculated the remainder in this case to be r = 203/4. Now, let’s see what happens when we evaluate a(x) for x equal to the zero of b(x), which is x = -3/2. We have

a(-3/2) = 203/4.

This turned out to be the same as the remainder we calculated using the division procedure once again.

Chinese Remainder Theorem

Under the assumption that the divisors are pairwise coprime, the Chinese remainder theorem asserts that if one knows the remainder of the Euclidean division of an integer n by multiple integers, one can uniquely determine the remainder of the division of n by the product of these numbers. The theory was first stated in the 3rd century AD by the Chinese mathematician Sunzi in Sunzi Suanjing.

The Chinese remainder theorem is commonly employed in large integer computing because it permits a computation for which abound on the size of the result to be replaced by numerous small integer computations. Every major ideal domain is true according to the Chinese remainder theorem (expressed in terms of congruences). With a definition incorporating ideals, it has been expanded to any commutative ring.

Remainder theorem and factor theorem

The remaining factor theorem is made up of two theorems that link a polynomial’s roots to its linear factors. The theorem is frequently used to factor polynomials without having to utilize long division. This gives us a powerful tool to factor polynomials, especially when used with the rational root theorem.

Remainder Theorem:

For a polynomial f(x), the remainder of f(x) upon division by x-c is f(c).

Factor Theorem:

Let f(x) be a polynomial such that f(c) =0 for some constant c. Then x-c is a factor of f(x). Conversely, if x-c is a factor of f(x), then f(c)=0.

Remainder Theorem Sums

Here are some sums of the Remainder Theorem.

Solve -4x^3+5x^2+8 by x+3

Then, Solve 4x^3+x^2+x-3 by x+5

Solve 4x^3-3x^2-8x+4 by x+1

Then, Solve 5x^3+x^2+x-8 by x+1

Solve 5x^3+x^2+x-9 by x+7

Then, Solve 5x^6-3x^3+8 by x+1

Solve 5x^6-3x^3+8 by x-1

Then, Solve x^2+5x+6 by x-1

Solve x^2-3x+3 by x-1

Then, Solve x^3+3x^2+8x+12 by x-1

Solve x^3+4x^2-8x-10 by x+1

Then, Solve x^3-7x^2+12x-10 by x+1

Solve x^3-x^2-44x+16 by x+1

Then, Solve x^4+8x^3+12x^2 by x+1

Solve x^4+8x^3+12x^2 by x+1

Then, Solve 3x^6+3x^4-3x^2+6 by x+1

Solve x^3-4x^2-9x+7 by x+1

Then, Solve x^2-2x+2 by x+1

Solve x^2-2x+3 by x+1

Then, Solve x^2-2x+4 by x+1

Solve x^2-3x+3 by x+4

Then, Solve x^2-3x+4 by x+2

Solve x^2-4x+2 by x+2

Then, Solve x^2-4x+4 by x-2

Solve x^2-4x+5 by x+2

Then, Solve x^2-2x+2 by x+8

Solve x^2-2x+3 by x+2

Then, Solve x^2-2x+4 by x+2

Solve x^2-2x+5 by x+2

Then, Solve x^2-3x+2 by x+2

Solve x^2-3x+3 by x+3

Then, Solve x^2-3x+4 by x+2

Solve x^2-3x+5 by x+2

Then, Solve x^2-4x+2 by x+6

Then, Solve x^2-4x+4 by x+2


What are the different methods of solving remainder when a polynomial expression is divided by a linear factor?

To find the remainder of a polynomial split by the linear factor, we have three techniques. Polynomial Long Division, Synthetic Division, and the Remainder Theorem are all options.

Is Factor Theorem and Remainder Theorem the same?

According to the factor theorem, if an is the zero of a polynomial p(x), then x-a is the factor of p(x), and vice versa. The remainder theorem states that the remainder of any polynomial p(x) divided by an x-a is equal to the f. (a).

Can you use the remainder theorem If the remainder is zero?

When a polynomial function is divided by its factor, the residual is zero. The remainder theorem can be used to any polynomial function, but the denominator must be a binomial of the type x-a.

What is the remainder for (2x^2 – 5x – 1) / (x – 3)?

Here, a = 3

f(x) = 2x^2 – 5x – 1

Then f(3) = 2 * 3^2 – 5(3) – 1

= 2 * 9 -15 – 1

= 2

What Is the Remainder Theorem in Math?

The remainder theorem allows us to calculate the remainder of any polynomial divided by a linear polynomial without actually performing the division algorithm steps.

The remainder is obtained by r = a when a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k. (k).

How do you Use the Remainder Theorem?

We can concentrate on the following procedures to use the remainder theorem:

  • Take note of the given polynomial.
  • Arrange the polynomial in order of increasing power.
  • Either perform the long division or by using the remainder theorem that is p(x) = (x-c)·q(x) + r(x) we can justify the answer.

Are the Factor Theorem and the Remainder Theorem Same?

The factor theorem states that if a polynomial a(x) has a value of 0 at x equal to any real number k, then the linear polynomial (x-k) is a factor of a(x), and we can write a(x) = (x-k) b(x), where b(x) is a polynomial of degree one less than a(x) (x).

What Is the Remainder Theorem Formula?

The general formula for remainder theorem is represented as, p(x) = (x-c)·q(x) + r(x)

Who Invented the Remainder Theorem?

The remainder theorem was created by Chinese mathematician Sun Zi. Qin Jiushao proved the complete remainder theorem in 1247.

What if the Remainder Is Zero?

If the remainder is zero, the factors of the provided equation are the remaining quotient and the divisor.

What Is the Use of the Factor Theorem?

The factor theorem is a mathematical technique for determining the components of a polynomial problem. Let’s pretend that f(x) is a polynomial and if f(k) = 0, then (x – k) is a factor of f. (x).

What Is the Use of Remainder Theorem Formula?

When a polynomial p(x) is divided by (ax + b), the remainder theorem formula is used to get the remainder. We can determine if (ax + b) is a factor of p(x) or not using the remainder theorem. (ax + b) is a factor of a polynomial p(x) if the remainder is 0, otherwise, it is not.

What Are the Applications of the Remainder Theorem Formula?

The principal application of the remainder theorem formula is the factor theorem. The remainder theorem is required to show the factor theorem. If the residual obtained by dividing p(x) by (x – r) is 0, then (x – r) is a factor of p, according to the factor theorem (x).

Some More Frequently Asked Questions

  • What does the Polynomial Remainder Theorem state?

The Polynomial Remainder Theorem states that when a polynomial f(x) is divided by a linear divisor (x – c), the remainder is equal to f(c).

  • How is the Remainder Theorem related to polynomial division?

The Remainder Theorem helps simplify polynomial division, as it allows us to find the remainder without performing long division when dividing by a linear factor.

  • Where can you find worksheets on the Polynomial Remainder Theorem?

Worksheets on the Polynomial Remainder Theorem can be found on various educational websites, such as Kuta Software, Math-Aids, or Teachers Pay Teachers.

  • Can you use a calculator for the Polynomial Division Remainder Theorem?

Yes, graphing calculators like the TI-84 or online tools like Wolfram Alpha can compute polynomial division and apply the Remainder Theorem.

  • How is the Remainder Theorem applied to polynomial functions?

The Remainder Theorem helps determine the remainder of a polynomial function without performing the actual division, which can simplify problem-solving and analysis.

  • What is the Chinese Remainder Theorem for polynomials?

The Chinese Remainder Theorem for polynomials is an extension of the number theory concept, stating that if you know the remainders when dividing a polynomial by multiple pairwise coprime polynomials, you can find a unique remainder modulo their product.

  • How does the Remainder Theorem relate to polynomial long division?

The Remainder Theorem helps bypass polynomial long division by directly finding the remainder when a polynomial is divided by a linear factor, saving time and effort.

  • How is the Polynomial Factor Remainder Theorem connected to the Remainder Theorem?

The Polynomial Factor Remainder Theorem is an application of the Remainder Theorem; it helps identify factors of a polynomial by finding the values of x that yield a remainder of zero.

  • What does the Polynomial Equation Remainder Theorem entail?

The Polynomial Equation Remainder Theorem involves using the Remainder Theorem to solve polynomial equations by finding the remainder and determining when the remainder equals zero, indicating a factor or a root.

  • Is there a calculator for the Polynomial Chinese Remainder Theorem?

Calculators for the Polynomial Chinese Remainder Theorem are not as common as basic polynomial division calculators, but specialized online tools and some computer algebra systems like Mathematics may support this feature.