between what consecutive integers are the zeros of F(x)=x^4-x^3+2x^2+5x-7?

Question

helder_edwin · Answer

try using a software to graph the function. it will give a hint where to look for.

anonymous · Answer

@helder_edwin I have no idea what im looking for lol or what to do ? can you help ?

whpalmer4 · Answer

First thing to realize is that these functions all have their "interesting stuff" (zeros) close to x = 0, because the end behavior is such that the function rapidly gets very large or small.

You can just pick an integer close to 0 (might start with -5, for example) and evaluate the function there.  Take note of the sign of the result.  Now move along the x axis by 1, and repeat the process.  Any time two adjacent values have differing signs, you know that there must be a zero between them.  The highest power of $x$ in this polynomial is 4, so you have 4 zeros, and they may not all be real zeros, you may have some complex conjugate zeros in the mix.  If you know Descartes' Rule of Signs, you can can see how many real roots to expect, and whether they are positive or negative.

You can also try the positive and negative factors of -7 as potential zeros.  If one or more of them turn out to be zeros, you can divide the entire polynomial by $(x-z_i)$ where $z_i$ is a zero and get a simpler polynomial which will have the same remaining zeros but will be easier to work with.