A Remainder Theorem in Algebra

Everyone wishes to learn a shortcut, including navigating paths or some other lengthy assignment. Finding out a quicker and more productive way to complete at the same endpoint makes us assume good since we’ve most likely recouped some time, work, and maybe money. Mathematics is served with these shortcuts, and one of the more valuable ones is the remainder theorem. It should be considered as the remainder theorem hardly works when a function is divided by a linear polynomial, and it should be in the form x + number, though it might not appear as, at least, in the first picture.

Let us check what the remainder theorem is? This algebraic theorem is expressed as follows:-

When a polynomial such as  (X) gets divided by a linear polynomial such as Y(X), which has 0 as its X = K, then the remainder is delivered as R = A(K). This algebraic theorem allows us to evaluate the remainder of the district of a linear polynomial by a polynomial without entirely reaching the points of the division algorithm.

The common procedure for this theorem is:-

This theorem is asserted as 

E[X] = (X-C)·q[X] + R[X].

Let us assume polynomials to verify the formula of the remainder theorem.

Let E[X] be any polynomial of substantial degree>/=1, and any real number is considered ‘S’. If E[X] is divided by the polynomial [ X– S], then the remainder is E(S). In other words, If H[X] as a polynomial is divided by X – S, then [R] as the remainder is provided by

H[X] = [X – S] q[X] + R, where the quotient is considered as q[X]  and R is consistent because of the extent of the remainder < the degree of the divisor [X – S].

Imposing

X = S, H(S) = (S – S)q(S) + R or H(S) = R

When H[X] as a polynomial is divided by X – S, we get R as a remainder = H(S) =  value of H[X] when X is S.

This remainder theorem regulates the proof that a polynomial is divisible once by its component to attain a smaller polynomial and a remainder of 0.

The objective of the remainder theorem

This algebraic theorem is particularly beneficial when it is paired with unnatural division. The unnatural division is an alternate procedure to shortly and handily divide polynomials rather than operating lengthy divisions. Furthermore, recall that the remainder is the number in the ground row in the last column on the right in the unnatural division. Therefore, rather than pushing a value in and obtaining the decree of systems, we can utilise unnatural division to analyse a polynomial for a provided value.

Another theorem is known as The Chinese remainder theorem.

The Chinese remainder theorem is the old theorem that provides the situations essential for multiple equations to have a coincidental integer outcome. This algebraic theorem has its lineage in the function of the third century AD. Sun Zi was a great Chinese mathematician, but Qin Jiushao first introduced the detailed theorem in 1247.

It deals with the following kind of problem. One is ordered to get a number that gives a remainder of zero when divided by five, a remainder six when divided by seven, and a remainder two when divided by twelve. The easiest explanation is three hundred and seventy. Point out that this explanation is not different since anything multiple of 5 × seven × 12 (= 420) can be expanded to it, and the conclusion will still figure out the question.

What is the usage of the Remainder theorem formulation?

The remainder theorem formulation is utilised to discover the remainder when a polynomial p(x) is divided by (ax + b). Obtaining the remainder theorem, we can specify the expression (ax + b) is a factor of p(x). If the remainder is zero, then (ax + b) is a factor of a polynomial p(x).

Conclusion

 The main application of the remainder theorem formula is the factor theorem. To verify the factor theorem, it explains that if the R that is obtained by dividing p(x) by (x – r) is zero, then (x – r) is a factor of p(x). If the remainder is 0, then the surviving quotient and the divisor are the factors of the provided expression.

The remainder theorem allows us to evaluate the remainder of the division of any polynomial by a linear polynomial without entirely reaching the points of the division algorithm. When a polynomial an (x) is divided by a linear polynomial b(x) whose 0 is x = k, the remainder is delivered as R = a(k).