Remainder Theorem

The remainder theorem is helpful to evaluate polynomials at a specific value of x. This remainder theorem is used to calculate the remainder when a polynomial is divided by another linear polynomial. When the divisor polynomial is linear you don’t need to perform the long division method, instead, you can simply use the remainder theorem. Let’s discuss the remainder theorem formula for better understanding. 

Remainder Theorem Formula

The remainder theorem states that when a polynomial p(x) is divided by a linear polynomial (x-a), then the remainder would be p(a). Or in better words when a polynomial p(x) is divided by a linear polynomial (ax+b), the remainder is p(-b/a). 

Note: Remainder theorem is further used to explain factor theorem. 

Remainder Theorem Formula:

 When p(x) is divided by (x-a)

Remainder =p(a)

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

Remainder =p(-b/a)

Remainder Theorem Proof

We can assume that f(x) is a polynomial with a degree greater than or equal to 1.

Let us suppose that when f(x) is divided by  linear polynomial p(x) = (x-a), the quotient is q(x) and the remainder is r(x).

Therefore , f(x) and p(x) are two polynomials such that the degree of f(x) ≥ degree of p(x) and p(x) ≠ 0 

We can hence identify polynomials q(x) and r(x) such that r(x) = 0 or degree of r(x) < degree of g(x).

Using division algorithm, we can derive that

f(x) = p(x) . q(x) + r(x) [dividend = divisor . quotient + remainder]

By placing values in the above equation, we get

 f(x) = (x-a) . q(x) + r(x) [ p(x) = x – a ]

Here, degree of p(x) = (x-a) is 1 and degree of r(x) < degree of (x-a)

Therefore,

Degree of r(x) = 0

This implies that r(x) is a constant, let us suppose as ‘ k ‘

So, for every real value of x, r(x) = k.

Therefore f(x) = ( x-a) . q(x) + k

If x = a,

then f(a) = (a-a) . q(a) + k = 0 + k = k

Therefore, the remainder when f(x) is divided by the linear polynomial (x-a) is f(a).

Application of Remainder Theorem

Suppose f(x) is a polynomial of degree ≥ 1 then

• The calculated remainder when f(x) is divided by ( x+a) is f(-a).
• The calculated remainder when f(x) is divided by ( ax +b) is f(-b/a).
• The calculated remainder when f(x) is divided by ( ax – b) is f(b/a) 

Remainder Theorem Questions With Solutions

Example1. Find the remainder when the polynomial p(x) = x4 + 2×3 – 4x – 3 is divided by (x – 3).

Since x – 3 is divisible by the polynomial, hence

x – 3 = 0 

⇒ x = 3

On applying remainder theorem,

When p(x) is divided by (x-a)

Remainder =p(a)

Therefore, Remainder = p(3)

= 34 + 2 (3)3 – 4(3) – 3

= 81 + 54 – 12 – 3

= 120

Thus, Remainder is = 120.

Example2. Determine whether (2x – 3) is a factor of p(x) = 2×3 + x2 + 4x -15.

2x – 3 = 0 

⇒ 2x = 3

⇒ x = 3/2

Putting the value of x in the remainder when p(x) is divided by (2x – 3). If the remainder is 0, (2x – 3) is a factor of p(x).

Remainder = p(3/2)

= 2(3/2)3 + (3/2)2 + 4(3/2) -15

= 27/4 + 9/4 + 6 – 15

= 0

Answer: (2x – 3) is a factor of p(x).

Example4. Find the remainder when 2×3 + 4×2 – 8x + 4 is divided by 2x + 1

Let us assume f(x) =2×3 + 4×2 – 8x + 4

Acc to question, divisor = 2x + 1

According to the remainder theorem, the required remainder is f(-1/2)

put x = -1/2 in above equation 

f(x) =2×3 + 4×2 – 8x + 4

f(-1/2) = 2(-1/2)3 + 4(-1/2)2 – 8(-1/2) + 4

= -(1/4) + 1 + 4 + 4 

= (-1+4+16+16)/4

= 39/4

Answer: Remainder = 39/4

Remainder Theorem Calculator

Various types of online remainder theorem calculators are available online. They take the dividend and divisor value from the user and provide the remainder within seconds. Let us know how to use these remainder theorem calculators.

The procedure to use such a remainder theorem calculator is

Step 1: Enter the numerator and denominator polynomial in the input field provided respectively.

Step 2: Now click the button “Divide” or “calculate” to get the output

Step 3: Finally, the quotient and remainder will be displayed.

Some of these Remainder theorem calculators also provide a detailed explanation of the process used for better understanding.

Conclusion

Discovering a more efficient and quicker route to the same destination makes you feel good because you’ve likely saved time, effort, and/or money. Mathematics is full of these kinds of shortcuts, and one of the most effective is the remainder theorem.