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

Jul 6, 2015

$3 b . \left(a + 1\right)$

#### Explanation:

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

Then,
With numbers : $3 \cdot 4 = 4 \cdot 3 = 12$
With letters : $a \cdot b = b \cdot a$

We can re-write your expression :

$2 a b + \textcolor{red}{b a} + 3 b = 2 a b + \textcolor{red}{a b} + 3 b$


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

$\to$It remains : $2 a b + a b$, it's 2 times the product of $a$ and $b$ added to the product of $a$ and $b$.
$\implies$This is the same result that if I multiply directly 3 times the product of $a b$

We can write : $\textcolor{b l u e}{2 a b + a b = 3 a b}$

Consequently our expression is now equal to :

$\textcolor{b l u e}{2 a b + a b} + 3 b = \textcolor{b l u e}{3 a b} + 3 b$



Last step : factorize $3 a b + 3 b$ !

After that, we have to find the common factor inside the addition :
$3 a b = \textcolor{red}{3} \cdot a \cdot \textcolor{g r e e n}{b}$ and $3 b = \textcolor{red}{3} \cdot \textcolor{g r e e n}{b}$

Then the common factor of $3 a b$ and $3 b$ is $\textcolor{red}{3} \cdot \textcolor{g r e e n}{b} = 3 b$ and so : $3 a b = 3 b \cdot a$ and $3 b = 3 b \cdot 1$ ( don't forget the 1-factor)

Therefore, the factorization of $3 a b + 3 b$ 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 !