Division Algorithm for Polynomials: Let p(x) and g(x) be two polynomials such that degree of p(x) ≥degree of g(x) and g(x) ≠ 0. Then there exists unique polynomials q(x) and r(x) such that p(x) = g (x) x q(x) + r(x)

Division of Polynomials

Let us consider the numbers 13 and 5. When 13 is divided by 5 what is the quotient and remainder.?

Yes, of course, the quotient is 2 and the remainder is3.We write 13 = (5×2)+3

Let us try.

From the above examples, we observe that the remainder is less than the divisor.

Can we say ? that when the remainder is 0, then the dividend is the multiple of the divisor?

Dividend = ( Divisor × Quotient ) + Remainder.

Is it possible to divide one polynomial by another ?

Of course, yes, and the way to do it is just the way similar to what you do with numbers!

Let us start with the division of a polynomial by a monomial.

Divide x3 -4x2 + 6x by x, where , x ≠ 0

Solution

We have

Let p(x) and g(x) be two polynomials such that degree of p(x) ≥degree of g(x) and g(x) ≠ 0. Then there exists unique polynomials q(x) and r(x) such that

p(x) = g (x) x q(x) + r(x) … (1)

where r(x) = 0 or degree of r(x) < degree of g(x).

The polynomial p(x) is the Dividend, g(x) is the Divisor, q(x) is the Quotient and r(x) is the Remainder. Now (1) can be written as

Dividend = ( Divisor × Quotient ) + Remainder.

If r(x) is zero, then we say p(x) is a multiple of g(x). In other words, g(x) divides p(x).

If it looks complicated, don’t worry ! it is important to know how to divide polynomials, and that comes easily with practice. The examples below will help you.

Find the quotient and the remainder when (5x2-7x+2) / (x-1)

*Solution*

Find quotient and the remainder when *f(x)* is divided by* g(x)*

*(i) f(x)* = (8*x3*–6*x*2+15*x*–7),* g(x)* = 2*x*+1. (ii) *f(x)* =* x*3* *+ 1,* g(x)* =* x*+1

*Solution*

If *x*4 –3*x*3 + 5*x*2 –7 is divided by *x*2 + *x* + 1 then find the quotient and the remainder.

*Solution*

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9th Maths : UNIT 3 : Algebra : Division of Polynomials | Explanation, Example Solved Problems | Algebra | Maths

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