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Q: Is it possible that the polynomial function doesn't have zeros?

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A quadratic polynomial must have zeros, though they may be complex numbers.A quadratic polynomial with no real zeros is one whose discriminant b2-4ac is negative. Such a polynomial has no special name.

The values of the variables which make the polynomial equal to zero

Either graph the polynomial on graph paper manually or on a graphing calculator. If it is a "y=" polynomial, then the zeroes are the points or point where the polynomial touches the x-axis. If it is an "x=" polynomial, then the zeroes are the points or point where the polynomial touches the y-axis. If it touches neither, then it has no zeroes.

If the cubic polynomial you are given does not have an obvious factorization, then you must use synthetic division. I'm sure wikipedia can tell you all about that.

In the general case, this is quite tricky. In high school, you learn some simple cases. If the polynomial is of degree 2, you can use the quadratic function. For higher degrees, in some specific cases you can use the methods taught in high school to factor the polynomial. As you might know, once the polynomial is completely factored, it is quite trivial to find the zeros. But in the general case, you need some iterative method, which is more appropriate for a computer. From Wikipedia, article "Polynomial": "Numerical approximations of roots of polynomial equations in one unknown is easily done on a computer by the Jenkins-Traub method, Laguerre's method, Durand-Kerner method or by some other root-finding algorithm." You can read about any of these methods for more information; but don't expect a formula where you just "plug in some numbers"; rather, those are iterative methods, that is, you need to repeat a certain calculation over and over until you get a root of a polynomial with the desired accuracy.

Related questions

The zeros of a polynomial represent the points at which the graph crosses (or touches) the x-axis.

Find All Possible Roots/Zeros Using the Rational Roots Test f(x)=x^4-81 ... If a polynomial function has integer coefficients, then every rational zero will ...

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.

by synthetic division and quadratic equation

no a plynomial can not have more zeros than the highest (degree) number of the function at leas that is what i was taught. double check the math.

A quadratic polynomial must have zeros, though they may be complex numbers.A quadratic polynomial with no real zeros is one whose discriminant b2-4ac is negative. Such a polynomial has no special name.

Since there are two zeros, we have: y = (x - (-2))(x - 7) y = (x + 2)(x - 7)

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.

Any multiple of X^2+X/2-1/2

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.

x3 + 4x2 - 25x - 100 = 0

Yes - but only if the domain is restricted. Normally the domain is the whole of the real numbers and over that domain it must have at least one real zero.

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