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.
