Remainder Theorem

The remainder theorem is a helpful rule in algebra that lets us find the remainder when a polynomial is divided by an expression like x - c.

We don’t have to do the full division process here. Instead of dividing, we can substitute the value c into the polynomial. The result that we get will be the remainder.

It’s an effective method that saves time and helps you check if a number is a root of the polynomial. 
 

The remainder theorem states that when a polynomial p(x) is divided by x - a, the remainder will be equal to the value of the polynomial at x = a. That is, r = p(a). 

Follow these steps to find the remainder:

Step 1: Identify the value of a from the divisor x - a.

Step 2: Substitute x = a in p(x)

Step 3: The result gives the remainder. 
 

The remainder obtained when a polynomial is divided by a linear polynomial can be determined directly using substitution. Evaluate the polynomial at the zero of the divisor.
 

For better understanding, consider the following example, of two polynomials

Dividend, \(p(x): 4x^3 + 5x^2 - 3x + 1\) 

Divisor: \(x - 2 \)

Step 1: Find the zero of the divisor 

\(x - 2 = 0  \)

\(x = 0  \)

Step 2: Apply the remainder theorem 
The Remainder Theorem states, 
Remainder = p(zero of divisor) \(= p(2) \)

Substitute \(x = 2 \) into \(p(x) \):

\(p(2) = 4(2)^3 + 5(2)^2 - 3(2) + 1\)

Step 3: Simplify further

\(4(8) + 5(4) - 3(2) + 1 \)

\(32 + 20 - 6 + 1 = 47 \) 
\(\)
So, the remainder is 47.
 

Now that we know what the remainder theorem is, let us learn the difference between the factor theorem and the remainder theorem.

Although the factor theorem is similar to the remainder theorem, they both serve a slightly different purpose. Let’s see how they differ in the table below:
 

What is The Remainder Theorem Standard and Proof? 

The remainder theorem states that if a polynomial f(x) is divided by x - c, then the remainder is f(c). This means that we don’t have to do long division; just place c into the polynomial, and the answer you get is the remainder of the division.

When p(x) is divided by (x - a): \(remainder = p(a) \)

or,

when p(x) is divided by (ax + b): \(remainder = p (-\frac {b}{a})\)

Furthermore, for different types of linear polynomials, we can extend the remainder theorem as given below:
 

• If p(x) is divided by x - a, the remainder is \(p(a) \; (\because \, x - a = 0 \Rightarrow x = a) \)
   
• When p(x) is divided by ax + b , the remainder is \(p\left(-\frac{b}{a}\right) \; (\because \, ax + b = 0 \Rightarrow x = -\frac{b}{a}) \)
   
• The remainder when p(x) is divided by ax - b is \(p\left(\frac{b}{a}\right) \; (\because \, ax - b = 0 \Rightarrow x = \frac{b}{a}) \)
   
• When p(x) is divided by bx - a is \(p\left(\frac{a}{b}\right) \; (\because \, bx - a = 0 \Rightarrow x = \frac{a}{b}) \)

Proof of Remainder Theorem

Assuming q(x) as the quotient and r as the remainder. Let us divide a polynomial p(x) by a linear polynomial (x - a).

According to the division algorithm,

\(Dividend = (divisor × quotient) + remainder\)

so, 

\(p(x)=(x−a)⋅q(x)+r\)

Substituting x = a, we get

\(p(a) = (a - a) ⋅ q(x) + r\)

\(p(a) = 0 ⋅ q(a) + r\)

\(p(a) = r \)

Hence, proved, \(remainder = p(a) \)

The tips and tricks given below will help students get a good command on the topic by providing efficient methods to work with the topic.
 

•  It is key to identify the zero of the divisor, before any calculations, always find the zero first.
   
• The direct substitution method is quicker and more efficient when compared to the long division method.
   
• Always check for negative divisors if the divisor is of the form \(ax + b\). Convert to zero by solving \(ax + b = 0 \), \(x = - \frac{b}{a}\)
   
• A quick way to verify if a number is a factor of the polynomials is: if p(a) = 0, then x - a is a factor.
   
• Practice with higher degree polynomials to improve speed and efficiency. 

The remainder theorem is an algebraic tool often used in real world situations such as:
 

• Checking divisibility – Businesses can use it to determine if quantities of items or payments can be evenly distributed without performing lengthy calculations.
   
• Coding and cryptography – The theorem aids in algorithms that detect errors in data transmission by evaluating polynomial-based codes.
   
• Engineering calculations – Engineers can quickly check values in polynomial models of circuits, beams, or fluid flows without full computation.
   
• Computer graphics – It is used in rendering curves and animations modeled by polynomials, helping identify specific points efficiently.
   
• Calendar calculations – Mathematicians and programmers use it to calculate recurring events or determine weekdays for given dates using polynomial formulas.