Division of polynomials

Division of polynomials involves two cases, the first one is simplification,which is reducing the fraction and the second one is long division.



here we have to divide the numerator (3x+6) by 3. For that we have to divide 3x and 6 separately by 3.

like 3x/3 and 6/3

When we divide so, in 3x/3 , 3 and 3 will become cancelled.In 6/3, we get 2.

Finally we get the answer (x+2)

Otherwise, we can do the above problem in a different way as following.

= 3(x+2)/3

= x+2 (Here 3 and 3 get cancelled)

Let us consider another example.

Simplify: [x(x-5) + 7(x-5)] / (x-5)

Here at the numerator, we have two terms are added. In those two terms we have a common term (x-5).

In order to simplify , let us factor out (x-5) at the numerator.
After factoring out the common term (x-5) , we have

= (x-5)(x+7)/(x-5)

here (x-5) at both numerator and denominator would get cancelled.

finally we get (x+7)

If division of polynomials involves more than a simple monomial or binomial, which we can't simplify easily, then we can use long division method. It is just like the method long division method

Divide: x²+10x+9 by x+1

First, let us set up the division.

Division of polynomials

Now let us divide the leading term of dividend by the leading term of the divisor.

Here, we have x² as leading term in the dividend and x as the leading term in the divisor. So we have to divide x² by x.

The result would be "x". Now this "x" should be taken as quotient and each term of the divisor to be multiplied by this "x". So we will get x² + 1x. Now this x² +1x to be subtracted from the dividend. We will get the result 9x + 9.

Now leading term of (9x+9) to be divided by the leading term of the divisor(x+1). So we will have 9x/x = 9. Now this 9 should be taken at the place of quotient and we have to multiply each term of the divisor by this 9. We will get 9x+9. Then we have to subtract this from the already existing (9x+9).

Finally we get the remainder zero and quotient (x+9)
These are the examples in the topic Division of polynomials.
By using the above way, we can divide any polynomial by any polynomial.

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