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Remainder Theorem Statement

Remainder Theorem states that for any polynomial p(x) with a degree greater than equal to 1 if divided by a linear polynomial of the form (x-a), then the remainder of the division is “r” and the value of ‘r’ is given as,

r = p(a)

Now we can easily find the remainder of the division without actually performing the division of the polynomial, to find the remainder of the polynomial when it is divided by a linear polynomial follow the steps discussed below,

  • Find the zero of the given linear polynomial (x – a) by setting it equal to zero as

x – a = 0 ⇒ x = a

  • Substitute this value in the given polynomial to get the required remainder as, p(a).

Using this statement we can find the remainder of the polynomials when they are divided by different linear polynomials such as,

  • When p(x) is divided by x – a, then the remainder is p(a), as x – a = 0 ⇒ x = a
  • When p(x) is divided by ax – b, then the remainder is p(b/a), as ax – b = 0 ⇒ x = b/a
  • When p(x) is divided by ax + b, then the remainder is p(-b/a), as ax + b = 0 ⇒ x = -b/a
  • When p(x) is divided by bx – a, then the remainder is p(a/b), as bx – a = 0 ⇒ x = a/b

Remainder Theorem

Remainder Theorem is the basic theorem used in mathematics which is used to find the remainder of any polynomial when it is divided by a linear polynomial. The remainder theorem works on the principle of Euclidean Division Lemma.

But the remainder theorem also has some limitations, i.e. it works only when a polynomial is divided by a linear polynomial, else it fails. The remainder Theorem is exclusively mentioned for Class 9 students. Now let’s learn about the Reminder theorem, its proof, and others in detail in this article.

Table of Content

  • What is the Remainder Theorem?
    • Remainder Theorem Definition
  • Remainder Theorem Formula
  • Remainder Theorem Statement
  • Remainder Theorem Proof
  • Dividing a Polynomial by a Non-Zero Polynomial
  • Remainder Theorem of Polynomial
  • Euler Remainder Theorem
  • Factor Theorem
  • Remainder Theorem and Factor Theorem
  • Remainder Theorem Examples

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