Does adding a term of a different magnitude that is multiplied by 0 change the degree of the function?

Question

Consider some quadratic function $y=ax^2 + bx + c$. Consider changing the function such that $z = 0x^3 + ax^2 + bx^2 + c$. Can it be said that $y = z$? Can it be said that $z$ is a cubic function, since there is a term to the power of $3$, or since that entire term evaluates to $0$, is it still quadratic?

I would think that since the two functions ought to have the same codomain and graph, they can be said to be equal, which therefore means that $z$ must be a quadratic function since $y$ certainly is.

Answer 1

The degree of a polynomial is defined as the highest power of the variable used with a non-zero coefficient. Both of $x$ and $y$ then have degree 2, so are said to be quadratic functions. Also $y=z$ as the difference of $y$ and $z$ is zero.

Answer 2

This is a question about language, and the answer may be dependent on context. If we're talking about $y$ and $z$ as functions, which you probably are in precalculus, then yes you can and should say $y = z$. They take the same inputs to the same outputs. In that context I would describe both as a quadratic function.

If we allow the leading coefficient to be $0$, then one could say that quadratic polynomials are a special ("degenerate") case of cubic polynomials. This way of thinking might be useful in other parts of math. At the end of the day this is about communicating clearly, so just explain what you mean and you will be correct :)