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

**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 x ^{3} -4x^{2 }+ 6x by x, where , x **

**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 (5x^{2}-7x+2) / (x-1)

*Solution*

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

*(i) f(x)* = (8*x ^{3}*–6

*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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