Question

How do you simplify `2ab+ba+3b`?

Answer 1

`3b.(a+1)`

Explanation:

Recall : In a multiplication, the order of the factors does not matter.

Then,
With numbers : `3*4 = 4*3 = 12`
With letters : `a*b = b * a`

We can re-write your expression :

`2ab+color(red)(ba)+3b = 2ab + color(red)(ab) + 3b`

Now forget for a moment the last term : `3b`

`->`It remains : `2ab + ab`, it's 2 times the product of `a` and `b` added to the product of `a` and `b`.
`=>`This is the same result that if I multiply directly 3 times the product of `ab`

We can write : `color(blue)(2ab + ab = 3ab)`

Consequently our expression is now equal to :

`color(blue)(2ab + ab) + 3b = color(blue)(3ab) + 3b`

Last step : factorize `3ab + 3b` !

After that, we have to find the common factor inside the addition :
`3ab=color(red)(3)*a*color(green)(b)` and `3b=color(red)(3)*color(green)(b)`

Then the common factor of `3ab` and `3b` is `color(red)(3)*color(green)(b)=3b` and so : `3ab=3b*a` and `3b=3b*1` ( don't forget the 1-factor)

Therefore, the factorization of `3ab + 3b` is :

`color(blue)(3b)⋅color(red)a color(green)+ color(blue)(3b)⋅color(red)1 = color(blue)(3b).(color(red)a color(green)+ color(red)1)`

And it's done, you have your simplified expression ! :)

You can profit of the factorized form to find roots !