Question

For the given function, determine consecutive values of X between which each real zero is located f (x)= -7x^4-3x^3-16x^2+16x+15

Answer 1

However, one could try to give integer values to x and see how the function behaves.

f(0) = +15
f(1) = +5
f(2) = -153
We see that there must be a zero (or three) between x=1 and x=2
going the other way
f(-1) = -21
there must be a zero (or three) between x=0 and x=-1

and so on

Using a calculator (and being careful with signs and powers of negative numbers) it should be easy enough to cover an interval from -10 to +10 for example.

Since this is a 4th degree polynomial, then when you have found 4 zeros, you can stop, as there cannot be more than that.

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The principle used is that of "continuity".

If the function has a positive value (above zero) at some point and a negative value at another point, then somewhere in between, it MUST pass through zero. It cannot avoid it, if it is a continuous function (and all polynomials are continuous).

With modern computers (very fast), it is an acceptable method to find approximations for the zeros (there are applications where all you need is an approximation).

For example, we have found a zero between 1 and 2.
We could try 1.5
then, depending, we continue with a number either bigger than 1.5 or smaller than 1.5, making the interval smaller and smaller, until we have an acceptable approximation