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    Division Algorithm for Polynomials

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6th class > > Division Algorithm for Polynomials

Division Algorithm for Polynomials

polynomials class X

You know that a cubic polynomial has at most three zeroes. However, if you are given only one zero, can you find the other two? For this, let us consider the cubic polynomial x3−3x2−x+3. If one of its zeroes is 1, then you know that x – 1 is a factor.

If x-1 is a factor then when you divide the polynomial by x-1 we will get some quotient and a remainder of .

So, you can divide x3−3x2−x+3 by x – 1, to get the quotient x2−2x−3.

But the quotient is a quadratic polynomial. And we know how to find factors of the quadratic polynomial.

From the figure above or finding factors by splitting the middle term we get the factors as and

So we have x3−3x2−x+3 = x−1x2−2x−3= x−1x+1x−3

So, all the three zeroes of the cubic polynomial are now known to you as , ,

In general, we can apply long division to divide a polynomial by a quadratic polynomial.

We can apply an algorithm which is similar to Euclid’s division algorithm.

This says 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).

This result is known as the Division Algorithm for polynomials