Q: What is the definition for the zero of a polynomial function?

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A value of the variable that makes the polynomial equal to zero (apex)

Assuming the polynomial is written in terms of "x": It means, what value must "x" have, for the polynomial to evaluate to zero? For example: f(x) = x2 - 5x + 6 has zeros for x = 2, and x = 3. That means that if you replace each "x" in the polynomial with 2, for example, the polynomial evaluates to zero.

set the values of the y equal to zero

Yes, a polynomial function is always continuous

Actually, it is easier to define a polynomial as a function of a variable, x, in the form:p(x) = anxn + an-1xn-1 + ... + a2x2 + a1x + a0 where n is an integer and the ai are constants. A non-polynomial, then, is any function of x which contains one or more terms in x which are not in the form akxk.

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A value of the variable that makes the polynomial equal to zero (apex)

That's the definition of a "rational function". You simply divide a polynomial by another polynomial. The result is called a "rational function".

A zero of a polynomial function - or of any function, for that matter - is a value of the independent variable (often called "x") for which the function evaluates to zero. In other words, a solution to the equation P(x) = 0. For example, if your polynomial is x2 - x, the corresponding equation is x2 - x = 0. Solutions to this equation - and thus, zeros to the polynomial - are x = 0, and x = 1.

Assuming the polynomial is written in terms of "x": It means, what value must "x" have, for the polynomial to evaluate to zero? For example: f(x) = x2 - 5x + 6 has zeros for x = 2, and x = 3. That means that if you replace each "x" in the polynomial with 2, for example, the polynomial evaluates to zero.

It is any function of the form ax3 + bx2 + cx +d where a is not zero.

true

set the values of the y equal to zero

the zeros of a function is/are the values of the variables in the function that makes/make the function zero. for example: In f(x) = x2 -7x + 10, the zeros of the function are 2 and 5 because these will make the function zero.

It is useful to know the linear factors of a polynomial because they give you the zeros of the polynomial. If (x-c) is one of the linear factors of a polynomial, then p(c)=0. Here the notation p(x) is used to denoted a polynomial function at p(c) means the value of that function when evaluated at c. Conversely, if d is a zero of the polynomial, then (x-d) is a factor.

Yes. A monomial is a zero-degree polynomial. Although the prefix poly means "several" the definition allows for any finite number of terms.

There are many things that can be said about a polynomial function if its fourth derivative is zero, but the main thing you can know about this function from this information is that its order is 3 or less. Consider an nth order polynomial with only positive exponents: axn + bxn-1 + ... + cx2 + dx + e As you derive this function, its derivatives will eventually be equal to zero. The number of derivatives that are nonzero before they all become zero can tell you what order the polynomial function was. Consider an example, y = x4. y = x4 y' = 4x3 y'' = 12x2 y''' = 24x y(4) = 24 y(5) = 0 The original polynomial was of order 4, and its derivatives were nonzero up until its fifth derivative. From this, you can generalize to say that any function whose fifth derivative is equal to zero is of order 4 or less. If the function was of higher order than 4, its derivatives would not become zero until later. If the function was of lower order than 4, its fifth derivative would still be zero, but it would not be the first zero-valued derivative. So this experimentation yielded a rule that the first zero-valued derivative is one greater than the order of the polynomial. Your problem states that some polynomial has a fourth derivative that is zero. Our working rule states that this polynomial can be of highest order 3. So, your polynomial can be, at most, of the form: y = ax3 + bx2 + cx + d Letting the constants a through d be any real number (including zero), this general form expresses any polynomial that will satisfy your condition.

Yes. Note that specifically, the sum might be a constant (just a number), or even zero, but it is convenient to include those in the definition of "polynomial".