I am studying linear regression. I want to ask is the linear word in multiple linear regression refers to the linear relationship between the target variable and each of the regression coefficients b_0, b_1,b_2, ..., b_n? Also in multiple linear regression there is a linearity assumption does this assumption refers to the linear relationship between the dependent variable and each independent variables which we confirm using scatter plot?
MLR equation that I am using:
$\hat y = b_0 + b_1x_1 + b_2x_2 + \dots + b_n x_n$
Also if this linearity assumption is not satisfied should I then try polynomial regression? And is this polynomial regression also a kind of linear regression where the independent variable is non-linearly related to dependent variable but the regression coefficients are linearly related to target variable?
I know I have asked too many doubts so please bear with me and be kind to clear my doubts.