Division algorithm for polynomials — lesson. Mathematics State Board, Class 9.

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If you divide

$\frac{5}{2}$ , you will obtain

$2$ as the quotient and

$1$ as the remainder.

Also here

$5$ is the dividend,

$2$ is the divisor,

$2$ is the quotient, and

$1$ is the remainder.

So we can write

$5 = (2 \times 2) + 1$

or Dividend

$=$ (Divisor

$×$ Quotient)

$+$ Remainder

Note that the remainder is always less than the divisor.

In algebra, long division polynomial is an algorithm for dividing a polynomial by a same or lower degree polynomial, a generally used version of the common arithmetic technique known as long division.

Let’s try to divide the two polynomials.

Example:

$\frac{p (x)}{q (x)}$ where

$p (x) = {2 x}^{3} + x^{2} + x$ &

$q (x) = x$ .

$(2x^3 + x^2 +x) ÷ x$

$=$

$\frac {2x^3}{x}+$

$\frac{x^2}{x}+$

$\frac{x}{x}$

$=$

$2x^2$

$+$

$x$

$+$

$1$ .

In here,

$(2x^3 + x^2 +x)$ is the dividend,

$x$ is the divisor,

$2x^2$

$+$

$x$

$+$

$1$ is the quotient and

$0$ is the remainder.

Note that the remainder is

$0$ as the polynomial

$q(x)$ divides the polynomial

$p (x)$ without any remainder.

Important!

In this case,

$x$ is common to each term of

${2 x}^{3} + x^{2} + x$ . So we can write

${2 x}^{3} + x^{2} + x$ as

$x ({2 x}^{2} + x + 1)$ .

We say that

$x$ and

${2 x}^{2} + x + 1$ are factors of

${2 x}^{3} + x^{2} + x$ , and

${2 x}^{3} + x^{2} + x$ is a multiple of

$x$ as well as a multiple of

${2 x}^{2} + x + 1$ .

Example:

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1. Division algorithm for polynomials

Theory: