what is the quotient when 2×3 + x + 3 is divided by x + 1? This is a topic that many people are looking for. bluevelvetrestaurant.com is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, bluevelvetrestaurant.com would like to introduce to you Why Quotient is Half when Divide by (2x – 1) in Synthetic Division. Following along are instructions in the video below:
m anil kumar and in this video. We will understand how to divide polynomials. We we have two techniques of dividing polynomials one we say long division kind of like and the other one is synthetic division.
Which we make a table like this so ill discuss both in this particular. Example. The question.
Here is divide 2. X. Cubed.
Plus. X. Square.
Plus. 5. X.
Minus. 1. By 2 x.
Minus. 1. So.
Well divide by 2 x. Minus. 1.
The polynomial is 2 x. Cubed. Plus x.
Square. Plus 5 x. Minus.
1. When you do synthetic division. You actually write down.
Only the coefficients so in this case youll be right to get 2 coefficient of x square is 1 and then we write. 5 and then minus 1 and divide by something which makes it 0 right. Which is half correct so in synthetic division is very short kind of in this much of space.
You can divide the reason for doing so is that in synthetic division. You may have to divide your quotient. Which you get here by 2 will understand why so at the end of the video.
So lets begin so we have a binomial here degree 1 dividing into degree 3. Trying. Degree 3.
So we expect degree 2. As a quotient x. Squared.
X. Cube. So we have to multiply by x square.
That gives us 2 x cube minus x square when you take away you get 2 x square bring down 5x then again 2 x 2 x. Square so that means we should multiply by x that gives us 2x squared minus x. 5x minus x.
When we just take away. We get 6x bring down minus 1. It can go 3 x.
Plus. 3. Multiplying.
3. By 2 x. Minus.
1. Gives. Us.
6x minus. 3. The remainder will be plus.
2. And you can now write. The statement.
Which is 2x. Cubed plus. X.
Squared. Plus. 5x minus 1.
Divided by 2x minus. 1. Is equal to the quotient.
Which is x square plus. X. Plus.
3. Plus remainder. 2.
Divided by 2x minus 1. Right. Now.
Lets do the right side using the same thing. But synthetic division. So to begin with we first equate 2x minus 1 to 0.
And we say 2x equals to 1 x. Is equals to 1 2. And thats why we get 1 2.
Here. So i hope you understand this step. Why we have 1 2.
Here. Now the steps involved are bring down the leading coefficient. Which is 2 times.
It with divisor. Which is 1 2. In this case.
It gives you 1. And then you add them up so 1 plus. 1 is 2.
Again multiply you get 1 add them up you get. 6 multiply you get 3 add them up so what you get 2. So you get exactly the same remainder.
Which you got earlier so that matches do you see that your remainder matches. However. The quotient here is 2 2.
6. 1 1. 3.
Do you see a relation here. This one is half of what you get there. What is the reason pause.
The video and think about it we actually divide by what we divided by x minus half. Do you see that we divided by not 2x minus 1. What we did was we wrote 2x minus 1.
As two common x minus. Half. And what we have taken care of is this part x minus.
How do you see that we still have to take care of 2. We have divided further by 2 to get our quotient correct. Thats the whole idea so to further divide by 2 to get the quotient.
This is one way to look at it and therefore you could say that 2x squared. I mean cube plus x. Square.
Plus. 5x minus 1. Is equal to half of this half of this so how come this will be 1 1.
3. Which gives you the same thing. Which is x squared plus x.
Plus. 3. Right times.
Im writing the same division statement in a different way times. 2x minus 1. Plus.
The remainder. 2. Is okay you could write like this and get the same result.
And i hope you have understood why we are multiplying. I rather dividing this by 2 or multiplying by 1 2. Correct.
So its very important to understand thats why i have taken them side by side you could also think like this that you have this quotient 2x. Squared. Plus.
2x. Plus. 6.
And that will be multiplied by x. Minus. 1.
2. X. How do you see that so that gives you half as a factor here right so we could write that as 1 2.
Times 2 x. Square. Plus 2 x.
Plus 6. Times 2 x. Minus 1.
And you see that factor. 1. 2.
That divides into half giving you same result. What we got here. So.
Thats an alternate way of looking at the same thing so i hope this thats clearly understood since many of my subscriber were asking why do we divide by two if we have a factor. Like 2x minus 1. So.
If the factor is e. X. Minus.
P. Youre divided by a remember that i hope you appreciate it feel free to post questions. Share my videos and subscribe to them thank you and all the best .
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