Features and polynomial regression

[Ex]
In preceding example of house's price, assuming that there are only two features the frontage and depth of the house in this model. And in this case, there are some relationship between these two features, so these two feature can transfer into one new feature which equal to frontage $\times$ frontage. This feature contain the information of two features, sometimes it might get a better model.

Here is the plot with size x (frontage $\times$ frontage) and price (y)
[Plot 1 ]

If we use a quadratic model $y = \theta_0 + \theta_1 x + \theta_2 x^2$, it's plot maybe like this
[Plot 2]

Although it fit y good in the beginning, but this model doesn't make sense since this curve will decrease gradually. Then we try a cubic model $y = \theta_0 + \theta_1 x + \theta_2 x^2 + \theta_3 x^3$, it's plot maybe like this
[Plot 3]

It'll be better than the model 1 cause it doesn't eventually decrease. When the model contain the high-order features, scaling features will become more important since the range of unit will increase very rapidly and if others feature are not on the similar scale it will have some trouble that we've mentioned before Feature scaling.

In the final, we can use some math method to convert our current feature and it will be helpful.