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If you divide , you will obtain as the quotient and as the remainder.
Also here is the dividend, is the divisor, is the quotient, and is the remainder.
So we can write
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:
where & .
\((2x^3 + x^2 +x) ÷ x\)
\(=\) \(\frac {2x^3}{x}+\) \(\frac{x^2}{x}+\)\(\frac{x}{x}\)
\(=\) \(2x^2\) \(+\) \(x\) \(+\) \(1\).
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.
(x)\) without any remainder.
Important!
In this case, is common to each term of . So we can write as .
We say that and are factors of , and is a multiple of as well as a multiple of .
Example:
Example:
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