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Smooth function has derivatives of all orders. Polynomials have derivatives of all orders, thus polynomials are smooth functions.
For example: f(x)=2x+3 => f'(x)=2 => f''(x)=0 => f'''(x)=0...
So all derivatives exist. (Derivative being zero is ok.)
Their continuity can be proven using the Weierstrass (epsilon-delta) definition.
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How polynomials and non polynomials are alike?
they have variable
What is a straight continuous arrangement of an infinite number of points?
That would be an infinitely long line. Or just called a line if you are talking about graphs
Can polynomials with the same graph have different roots?
False! If the graph is exactly the same, then the x-intercepts will be the same which implies the roots are them same. However, you can have the same roots and different graphs. So while the first statement is true, the converse if not.
What are the operation of polynomials?
Adding and subtracting polynomials is simply the adding and subtracting of their like terms.
Can the sum of three polynomials again be a polynomial?
The sum of two polynomials is always a polynomial. Therefore, it follows that the sum of more than two polynomials is also a polynomial.
