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Sunday, July 3, 2022

What is Remainder Theorem? A Quick and Simple Introduction

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When a polynomial is divided by a linear polynomial, the remainder theorem is used to get the remainder. The remainder is the number of items left over after a particular number of items are split into groups with an equal number of items in each group. After division, it is anything that “remains.” Let’s have a look at the remainder theorem.

The remaining theorem goes like this: The remainder is obtained by r = a (k)when a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k. The remainder theorem allows us to determine the remainder of any polynomial divided by a linear polynomial without actually performing the division algorithm steps.

The remainder theorem is used in calculus to find the remainder of a division problem, where one function is divided by another, with rational functions as divisors. Read more about 5 Qualities of Good Languages Tutors.

The formula for the Remainder Theorem:

p(x) = (x-c)q(x) + r(x) is the generic formula for the remainder theorem. To show the remainder theorem formula, we’ll use polynomials.

One version of the theorem states that if f(x) and g(x) are continuous on [a, b] and g(x) does not equal 0 outside an open interval (a ,b), then for any c between a and b there exists d such that:

f (x) = (g(x)) + d (c − a)(b − c)(x − c).

The theorem is useful because it allows one to compute the remainder of f(x) divided by g(x) without actually performing the division algorithm.

To use this theorem, you need to know the zero of function g(x). To find out more on how to find these zeros, follow this link. Once you know what zero for function g(x) is, then you can substitute that into f(x) = ((g(X)) + d (c -a) (b -c) (X – c)). Then solve for X. The answer will be where your remainder is.

This theorem is used to find the remainder of a polynomial divided by another function. It is necessary because some functions don’t have X as one of their possible zeros, and therefore cannot be factored into smaller pieces. To learn more about how to find these zeros, please follow this link.

The remainder theorem can also be used with rational functions as divisors, though we first require the following lemma:

If p(x) and q(x) are polynomials such that q (c) = 0 for all c in an open interval containing a and b, then there exists some expression involving only basic arithmetic operations which equal 1 on the interval (a,b ) and equals 0 on the interval (c,b ).

Types of Remainder:

  • Negative Remainder Concept:

The remainder cannot be negative by definition. However, you might presume that in some circumstances for your convenience. A negative remainder, on the other hand, indicates that to determine the genuine remainder, you must add the divisor to the negative remainder.

Remainders’ Cyclicity:

Remains have a trait called cyclicity, which causes them to repeat themselves beyond a certain point.

Uses:

– This theorem can be used to find the remainder of a polynomial divided by another function.

– This theorem is also useful when finding the zeros on functions that cannot easily be factored into smaller pieces.

– The zero is where f(x) = ((g(X)) + d (c -a) (b -c) (X – c)), then solve for X to get one’s remainder.

Important Notes:

  • The remainder is obtained by r = a(k)when a polynomial a(x) is divided by a linear polynomial b(x) whose zero is x = k.
  • p(x) = (x-c)q(x) + r(x) is the remainder theorem formula.
  • Dividend = (Divisor * Quotient) + Remainder is the basic formula for determining the division.
  • The sum of five whole numbers in a row is always divisible by five.
  • When one divides any odd number by 8, they will get 1 as the remainder.
  • Any three consecutive natural integers have a product divisible by eight.
  • The product of any nine consecutive integers has a unit digit of zero
  • 10n-7 is divisible by 3 for any natural integer n.
  • Every three-digit number with the same digits is always divisible by 37.
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