Remainder Theorem

The remainder theorem states that if a polynomial p(x) is divided by a linear polynomial (x - a), the remainder of the division is simply p(a). This theorem makes it easy to find the remainder without performing the full long division process. Since the degree of a remainder is always one less than the degree of the divisor, dividing a polynomial by a linear polynomial of degree 1 will always result in a constant remainder that is degree 0.

According to the remainder theorem, when a polynomial p(x) of degree greater than or equal to 1 is divided by a linear polynomial x - a, the remainder is obtained by simply evaluating the polynomial at x = a. In other words, the remainder is r = p(a).


To find the remainder using the remainder theorem.
 

Step 1: First, find the zero of the linear divisor by equating it to zero. That is, set x - a = 0, which gives x = a.

Step 2: Next, substitute this value of x into polynomial p(x). The resulting value is the remainder.

This method forms the basis of many online tools, such as a remainder theorem calculator, and is closely related to the quotient-remainder theorem, which deals with both the quotient and the remainder obtained during polynomial division.
The remainder theorem can also be applied to other types of linear divisors, as shown.

• When p(x) is divided by x - a, the remainder is p(a) since x - a = 0, that is, x = a.
   
• When p(x) is divided by ax + b, the remainder is p(\(\frac{-b}{a}\)) since ax + b = 0, that is, x = \(\frac{-b}{a}\).
   
• When p(x) is divided by ax - b, the remainder is p(\(\frac{b}{a}\)) since, ax - b = 0 that is x = \(\frac{b}{a}\).
   
• When p(x) is divided by bx - a, the remainder is p(\(\frac{a}{b}\)) since, bx - a = 0 that is x = \(\frac{a}{b}\).

The remainder and factor theorem, also known as the factor and remainder theorem, states that if the remainder is zero, then the divisor is a factor of the polynomial. This concept is widely used in polynomial factorization, verifying factors, and understanding the relationships among factors, quotients, and remainders.

Assume that when a polynomial p(x) is divided by a linear polynomial (x - a), the quotient is q(x) and the remainder is a constant r. According to the division algorithm,

Dividend = (Divisor × Quotient) + Remainder

So, we can write it as,

\(p(x) = (x - a) q(x) + r\)

Now, substitute x = a in the above expression:

\(p(a) = (a - a) q(a) + r\)

\(p(a) = 0.q(a) + r\)

\(p(a) = r\)

Thus, the remainder obtained when p(x) is divided by (x - a) is equal to p(a).

Hence, the remainder theorem is proved.

The remainder can be easily determined when a polynomial is divided by a linear polynomial. Instead of performing lengthy calculations, we substitute the zero of the linear divisor into the given polynomial. This is the basic idea of the remainder theorem. For example,

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

Divisor, x + 1

We will find the remainder in two ways.
 

• Using the long division method
• Using the remainder theorem.

Let us check whether both methods give the same result.

Using polynomial long division:

After dividing \(4x^3-3x^2+5x-1\) by x + 1, the remainder obtained is r = 5.

Using the remainder theorem:

First, find the zero of the divisor:

x + 1=0

x = -1

Now, substitute x = -1 into the polynomial:

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

 = -4 -3 -5 -1

 = -13

So, the reminder obtained using the remainder theorem is -13.
Since the remainder found using the remainder theorem matches the remainder obtained through long division, the remainder theorem provides a quick and effective way to see the remainder when a polynomial is divided by a linear polynomial.

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:
 

• The remainder theorem states that if a polynomial p(x) is divided by a linear polynomial whose zero is x = k, the remainder of the division is p(k).
   
• In polynomial division, the relationship is given by. Dividend = (Divisor × Quotient) + Remainder.
   
• The remainder theorem is valid only for linear divisors.
   
• It cannot be applied if the divisor is not linear.
   
• The theorem is used to find the remainder only and does not provide the quotient.

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. 
   
• Parents should guide children to carefully handle negative signs when the divisor is in the form ax + b.
   
• Children can learn that the direct substitution method is quicker and more efficient than long division.

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.