Division Algorithm for Polynomials

Let us study what is meant by Division Algorithm for Polynomials,
Let us consider the cubic polynomial
x3 – 3x2 – x + 3. If we tell you that one of its zeroes is 1, then you know that x – 1 is
a factor of x3 – 3x2 – x + 3. So, you can divide x3 – 3x2 – x + 3 by x – 1, to get the quotient x2 – 2x – 3.
Next, you could get the factors of x2 – 2x – 3, by splitting the middle term, as
(x + 1)(x – 3). This would give you
x3 – 3x2 – x + 3 = (x – 1)(x2 – 2x – 3)
= (x – 1)(x + 1)(x – 3)
So, all the three zeroes of the cubic polynomial are now known to you as
1, – 1, 3.
Let us discuss the method of dividing one polynomial by another in some detail.
Before noting the steps formally, consider an example.
Example : Divide 2x2 + 3x + 1 by x + 2.
Solution : Note that we stop the division process when
either the remainder is zero or its degree is less than the
degree of the divisor. So, here the quotient is 2x – 1 and
the remainder is 3. Also,
(2x – 1)(x + 2) + 3 = 2x2 + 3x – 2 + 3 = 2x2 + 3x + 1
i.e., 2x2 + 3x + 1 = (x + 2)(2x – 1) + 3
Therefore, Dividend = Divisor × Quotient + Remainder

Let us now extend this process to divide a polynomial by a quadratic polynomial.