DIVIDING POLYNOMIALS WITH MISSING TERMS

When we want to divide a given polynomial by another polynomial, first we have to write the dividend inside the long division sign from highest degree to lowest degree.

For example, the highest degree of the polynomial is 3, then the next term of the dividend must be the square term and so on. In this, if we don't have square term, we have to write 0 instead of this and we can write the next terms.

Let us see some example problems based on the above concept.

Example 1 :

Do the following division : 

(a3 + 8a - 24) ÷ (a - 2)

Solution :

The degree of the given polynomial is 3 then we must have square term. But here we don't have square term, so we have to replace it by 0.

Step 1 :

In the first step, we have to divide the first term of the dividend by the first term of the divisor.

If we divide a3 by a, we will get a2. We have to write a2 at the top and multiply each term of the dividend by a2.

a2(a - 2)  =  a3 -  2a2

By subtracting a3 -  2a2 from (a3 + 8a - 24), we get 

2a2 + 8a + 24

Step 2 :

Now we have to divide 2a2 by a, so we will get 2a.

  • By multiplying this 2a by (a - 2), we get 2a2 - 4a.
  • By subtracting 2a2- 4a from 2a2 - 8a - 24, we get 12a - 24

Step 3 :

Divide 12a by a, so we get 12.

  • By multiplying this 12 by (a - 2), we get 12a - 24.
  • By subtracting 12a - 24 from 12a - 24, we get 0.

So, 

Quotient  =  a2 + 2a + 12

Remainder  =  0

Example 2 :

Do the following division :

(4x3 + 2x2 - 5)  ÷  2x

Solution :

The degree of the given polynomial is 3. But here we don't have x term, so we have to replace it by 0.

Step 1 :

In the first step, we have to divide the first term of the dividend by the first term of the divisor.

If we divide 4x3 by 2x, we will get 2x2.



  • Write this 2xat the top.
  • Multiply 2xby 2x.  So we get 4x3.
  • Subtract 4x3 from  4x3 + 2x2.

Step 2 :

Now we have to divide the first term of the dividend 2x2 by 2x, so we get x.

  • By multiplying x by 2x, we get 2x2.
  • Subtracting 2x2 form 2x2 we get 0
  • Bring down the next term, we get -5

So, 

Quotient  =  2x2 + x

Remainder  =  -5

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