# A detailed procedure for the synthetic division when the divisor is a second-degree polynomial

You most probably have learned the synthetic division method for the polynomial when divided by . But, what if the divisor is a second-degree polynomial? Now, the method you were trained will not work.

In this post, I will try to explain a bit more detail about the method which will work for our proposed problem. Let me start by defining a general expression for the polynomial as . Lets say, our divisor will be and reminder be . I must say, when you divide a polynomial of degree by another polynomial with degree then, the quotient will have a degree of . This means, in our case, we have a divisor with degree . So, our quotient will look like this .

If we use the Euclidean division algorithm which says “if you divide integers then, it produces a quotient and a remainder smaller than the divisor”. For our case,

.

Now, we collect the terms so that we can form a polynomial. i.e.

It looks like we can compare with the polynomial as

and .

If we make supposition and then, above compare becomes

and .

At last, you will know why I made this supposition!

Have you notice there is no ? This means . This is because the degree of quotient is . So we can say, must equal to . i.e.

.

In most of the case, a remainder is simply a number (i.e. ) but, if the remainder is a polynomial then, we need to write . This happens if . I would be very happy if you could find it by yourself why we can write .

Now, we have all the ingredients to make a method that works for us. Thus, the method looks like this:

. The red line shows . This is our initial visualization.

In this figure, the blue line shows is multiplied by and the green line shows is multiplied by . This means, always multiply one term further than . All the elements of the fourth column are then added and equal to .

Similarly,

. The last figure is our final visual method for the synthetic division when the divisor is second-degree polynomial.

Before finishing this post, isn’t it great if we can implement this visual method in any one related example? So, our example goes like this: “Find the quotient and remainder when is divided by ?”

Our solution starts as and . Now, by using our visual method, we can have

. Hence, the quotient is . Since the sum of the second last column is not equal to zero so, the remainder becomes . I must say, when you multiply with second last column’s last row then, you should multiply it with , not .

**Quick Question for You**

Lastly, if you could write a program (in any programming language except maths 😃) to implement this method then, please feel free to share with all of us via comment.

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If you guys have some questions, comments, or insults then, please don’t hesitate to shot me an email or comment below.

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