Remainder Theorem

The Remainder Theorem is an algebraic concept that allows us to quickly determine the remainder when a polynomial is divided by a linear expression of the form (x − a), without performing a long division.

Statement:

Instead of actually dividing the polynomial, we can just substitute x = ai nto the polynomial to get the remainder.

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

Given,

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

Using long division:

• Quotient = q(x) =2x² - x + 6
• Remainder = r(x) = -7

Now, let’s simplify factorization using the Remainder Theorem.

First, find the zero of the divisor:

• x + 2 = 0
• x  = -2

According to Remainder Theorem, substituting x = -2 in p(x),

Now, Remainder for p(-2) = -7

Thus, Remainder Theorem is verified.

Alternate Method

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

p(x) = (x−a)*q(x)+r, where q(x) is the quotient and r is the remainder.

If we substitute x = a in the above equation, we get

p(a) = r

Thus, the remainder is equal to the value of the polynomial at x=a

Similarly, for the divisor (x+2),

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

Putting x = −2 , we get

p(−2) = r

Hence, the remainder is p(−2)
Hence, proved.

Applications

• Simplifying polynomial division: The theorem eliminates the need for long division in certain cases.
• Factoring polynomials: Using the theorem with the Factor Theorem allows for easier factoring.
• Solving polynomial equations: It helps in checking whether a given number is a root of the polynomial.
• Signal processing: In engineering, polynomials are used to model signals, and the Remainder Theorem aids in simplifying these models.

Practice Questions on Remainder Theorem