Division of polynomials involves two cases, the first one is simplification,which is reducing the fraction and the second one is long division.
Simplify:
(3x+6)/3
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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