The division algorithm says when a number 'a' is divided by a number 'b' gives the quotient to be 'q' and the remainder to be 'r' then a = bq + r where 0 ≤ r < b. This is also known as "Euclid's division lemma". The division algorithm can be represented in simple words as follows:
- Dividend = Divisor × Quotient + Remainder
Let us just verify the division algorithm for some numbers. We know that when 59 is divided by 7, the quotient is 8 and the remainder is 3. Here,
- dividend = 59
- divisor = 7
- quotient = 8
- remainder = 3
- Verification of division algorithm:
Dividend = Divisor × Quotient + Remainder
59 = 7 × 8 + 3
59 = 56 + 3
59 = 59
Hence, the division algorithm is verified.
Division Algorithm for Polynomials Statement:
The division algorithm for polynomials states that, if p(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that
p(x) = g(x) × q(x) + r(x)
where r(x) = 0 or degree of r(x) < degree of g(x).
Here,
p(x) represents the dividend polynomial
g(x) represents the divisor polynomial
q(x) represents the quotient polynomial
r(x) represents the remainder polynomial
