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Remainder Theorem

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Remainder Theorem is the basic theorem used in mathematics which is used to find the remainder of any polynomial when it is divided by a linear polynomial. Suppose for any given polynomial f(x) if it is divided by a linear polynomial (x-a) then its reminder is always f(a).

The remainder theorem works on the principle of Euclidean Division Theorem. Remainder Theorem helps us to find the remainder of a polynomial when it is divided by a linear polynomial, it is used to find the factors of any polynomial, etc. But the remainder theorem also has some limitations, i.e. it works only when a polynomial is divided by a linear polynomial, else it fails. Remainder Theorem is exclusively mentioned for Class 9 students.

Now let’s learn about the Reminder theorem, its proof, and others in detail in this article.

Remainder Theorem Definition

Remainder Theorem states that for any polynomial whose degree is greater than or equal to 1 if it is divided by a linear polynomial p(x) such that p(x) = (x – a), the remainder of this division is always equal to p(a). This theorem is very helpful in finding the remainder of the polynomial without actually performing the division. The remainder theorem can be easily expressed as, for any polynomial p(x) { degree greater than or equal to 1} if divided by any linear polynomial s(x) = x-a the reminder is always p(a) such that, 

p(x)/s(x) = q(x) + r(x)

where,

  • q(x) is Quotient
  • r(x) is Remainder which is equal to r(a)

Remainder Theorem Formula

For any polynomial function f (x) when divided by the linear polynomial (x-a) then the remainder is always equal to f (a). 

f (x) = (x-a) × q (x) + r (x)

where r(x) is the remainder of the polynomial

The image given below express the remainder theorem.

Remainder Theorem Formula

Remainder Theorem Statement

Remainder Theorem states that for any polynomial p(x) with a degree greater than equal to 1 if divided by a linear polynomial of the form (x-a), then the remainder of the division is “r” and the value of ‘r’ is given as,

r = p(a)

Now we can easily find the remainder of the division without actually performing the division of the polynomial, to find the remainder of the polynomial when it is divided by a linear polynomial follow the steps discussed below,

  • Find the zero of the given linear polynomial (x – a) by setting it equal to zero as

x – a = 0 &#x21d2 x = a

  • Substitute this value in the given polynomial to get the required remainder as, p(a).

Using this statement we can find the remainder of the polynomials when they are divided by different linear polynomials such as,

  • When p(x) is divided by x – a, then the remainder is p(a), as x – a = 0 &#x21d2 x = a
  • When p(x) is divided by ax – b, then the remainder is p(b/a), as ax – b = 0 &#x21d2 x = b/a
  • When p(x) is divided by ax + b, then the remainder is p(-b/a), as ax + b = 0 &#x21d2 x = -b/a
  • When p(x) is divided by bx – a, then the remainder is p(a/b), as bx – a = 0 &#x21d2 x = a/b

Remainder Theorem Proof

For any polynomial g(x) the remainder theorem can easily be proved as,

Let g(x) be a polynomial with a degree of 1 or greater than 1. Suppose that when g(x) is divided by (x – b), the quotient is q(x) and the remainder is r(x), i.e.

g(x) = (x &#x2013 b) q(x) + r(x)

Since the degree of x – b is 1 and that of r(x) is less than that of x – b, the degree of r(x) = 0. This means that r(x) is constant. Thus, we can write the above equation as 

 g(x) = (x &#x2013 b) q(x) + r

In particular, if x = b, then the equation becomes,

p(b) = (b – b) q(b) + r 

p(b) = (0) q(b) + r

p(b) = r

Hence, Proved.

Dividing a Polynomial by a Non-Zero Polynomial

For dividing a polynomial with a non-zero polynomial use the steps given below,

Step 1: Arrange both the polynomial dividend and divisor in the descending order of their degree.

Step 2: The first term of the dividend is divided by the first term of the divisor to produce the quotient,

Step 3: After subtracting the first term arrange the remainder and copy the second term of the dividend and start dividing the remaining term as discussed above.

Step 4: Repeat these steps until the degree of the dividend polynomial is less than the divisor. In the end, only the remainder is left.

Remainder Theorem of Polynomial

How to use the Remainder theorem? can easily be explained with the help of the example given below.

Example: Divide 2x3 + 3x2 + 4x + 5 by x + 2

Solution:

Given,

Dividend = p(x) = 2x3 + 3x2 + 4x + 5

Divisor = s(x) = (x + 2)

Division of Polynomial

Here, 

Quotient = q(x) =2x2 – x + 6

Remainder = r(x) = -7

Verification:

Given,

Divisor = (x + 2)

Let x + 2 = 0

x  = -2

According to Remainder Theorem, substituting x = -1 in p(x),

remainder Theorem Verification

Now, Remainder  = p(-2)

Thus, Remainder Theorem is verified.

Alternate Method

We know that any polynomial p(x) can easily be written as,

p(x) = (x &#x2013 a)·q(x) + r

Taking x = a in the above equation we get only the remainder,

similarly, for any p(x)

p(x) = (x + 2).q(x) + r

if we put x = -2 we get the remainder of the above equation.

Thus, p(-2) is the remainder.

Hence proved.

Euler Remainder Theorem

For any two co-prime positive integers n and X, Euler&#x2019s theorem  states that,

Xφ(n) = 1 (mod n)

where φ(n) is called Euler&#x2019s function and its value is given as,

φ(n) = n (1-1/a)×(1-1/b)(1-1/c)

where 

  • n is a natural number
  • n = ap × bq × cr
  • a, b, c are prime factors of n
  • p, q, and r are positive integers.

Example: Find the Euler function of 21

Solution: 

Factors of 21 are,

21 = 7×3

φ(21) = 21 (1 – 1/7)(1 – 1/3)

          = 21 × 6/7 × 2/3

          = 12

Thus, the Euler function of 21 is 12

Factor Theorem

Factor Theorem is also the basic theorem of mathematics which is considered the special case of the remainder theorem. This is generally used the find roots of polynomial equations. 

This theorem states that for any polynomial p(x) if p(a) = 0 then (x – a) is the factor of the polynomial p(x). 

Differences between the Remainder Theorem and Factor Theorem

Remainder Theorem and Factor Theorem are the basic theorems in mathematics and the basic difference between them is stated in the table given below:

Remainder TheoremFactor Theorem
According to, Remainder Theorem for any polynomial p(x) when divided by x – a the remainder is p(a).According to Factor Theorem if (x – a) is a factor of p(x) then this is true only if f(a) = 0.
Remainder Theorem helps us to find the remainder of the polynomial without actually dividing it.Factor theorem helps us to find the factor of the given polynomial.

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Remainder Theorem Example

Example 1: Find the remainder when p(x) = x4 – x3 + x2 – 2x + 1 is divided by g(x) = x – 2.

Solution:

Given,

p(x) = x4 – x3 + x2 – 2x + 1

g(x) = x-2

Using Remainder theorem,

p(2) = (2)4 – (2)3 + (2)2 + 2(2) + 1 

       = 17

Thus, the remainder when p(x) is divided by g(x) then we get remainder as, 17

Example 2: Find the root of the polynomial x2 – 5x + 4

Solution: 

We know that if for any p(x) we get p(a) = 0, then x-a is the factor of p(x) or a is the root of equation.

Given,

f(x) = x2 – 5x + 4

By hit and try method.

f(4) = 42 – 5(4) + 4 

f(4) = 20 – 20 

       = 0

So, (x – 4) must be a factor of x2 – 5x + 4 

Example 3: Find the remainder when t3 – 2t2 + 4t + 5 is divided by t – 1.

Solution: 

Given,

p(t) = t3 – 2t2 + 4t + 5

g(t) = t – 1

Using Remainder theorem,

g(1) = (1)3 – 2(1)2 + 4 + 5 

        = 8

By the Remainder Theorem, 8 is the remainder when t3 – 2t2 + 4t + 5 is divided by t – 1

Example 4: Find the remainder when x3 x2 + 2 is divided by x – 2.

Solution: 

Given,

p(x) = x3 x2 + 2

g(x) = x-2 = 0

x = 2

Using Remainder theorem,

g(2) = (2)3 – (2)2 + 2

        = 6

By the Remainder Theorem, 6 is the remainder when x3 – x2 + 2 is divided by x – 2

Example 5: Find the root of the polynomial 3x2 – 7x + 2

Solution: 

We know that if for any p(x) we get p(a) = 0, then x-a is the factor of p(x) or a is the root of equation.

Given,

f(x) = 3x2 – 7x + 2

By hit and try method.

f(2) = 3(2)2 – 7(2) + 2 

f(2) = 12 -14 + 2

       = 0

So, (x – 2) must be a factor of 3x2 – 7x + 2 and 2 is the root of 3x2 – 7x + 2

FAQs on Remainder Theorem

1. What is Remainder Theorem?

Answer:

According to Remainder Theorem, for any polynomial p(x) (whose degree is greater than equal to 1) when divided by the linear polynomial (x – a), the remainder is always p(a). 

2. Who Invented Remainder Theorem?

Answer:

The credit for inventing remainder theorem goes to Chinese mathematician Sun Zi and the it was completed by Qin Jiushao in 1247.

3. What is the Use of Factor Theorem?

Answer:

Factor theorem is used to find the factors of the given polynomial. For any polynomial f(x), x-a is the factor of the polynomial f(x) only when, f(a) is zero.

4. What are Applications of Remainder Theorem?

Answer:

Remainder theorem is widely used to find the remainder of the polynomial without actually performing the long division and remainder theorem along with factor theorem is widely used to solve the polynomial equation.

5. What is Remainder Theorem Formula?

Answer:

For the polynomial p(x) when divided by the linear polynomial (ax+b) the Remainder Theorem Formula is 

p(x) / (ax + b)

Remainder = p (-b/a)



Last Updated : 28 Nov, 2023
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