Timeless Theorems of Mathematics/Polynomial Remainder Theorem

The Polynomial Remainder Theorem is an application of Euclidean division of polynomials. It is one of the most fundamental and popular theorems of Algebra. It states that the remainder of the division of a polynomial by a linear polynomial is equal to . Examples Example 1 Show that the remainder of the division of a polynomial by a linear polynomial is equal to . Solution : Divide by like the following one. x - 1 ) x^2 - 2x + 2 ( x - 1 x^2 - x ------------ - x + 2 - x + 1 ------------ 1 As, , thus the remainder is equal to . Example 2 Show that the remainder of the division of a polynomial by a linear polynomial is equal to . Solution : Divide by like the following one. x-m ) ax^2+bx+c ( ax+am+b ax^2-amx ------------------ amx+bx+c amx -am^2 ------------------ bx+c+am^2 bx-bm ------------------ am^2+bm+c As, , thus the remainder is equal to . Proof Proposition If is a polynomial of a positive degree and is any definite number, the remainder of the division of by will be Proof The remainder of the division of a polynomial of a positive degree by is either 0 or a non-zero constant. Let the remainder is and the quotient is . Then, for every value of , · Putting in the equation , we get · . Thus, the remainder of ÷ is equal to .