How many real roots does the polynomial f(x)=x^7+5x+12 have?

Question

How many real roots does the polynomial f(x)=x⁷+5x+12 have?

Answer 1

Steven G is partly correct... because this is an odd function (one end of the graph goes down while the other end goes up), there is at least 1, and as many as 7, real roots (because the exponent on the leading term is 7). The trick is finding them. You might start by graphing the function to see how many times it crosses the x-axis. This is how many real roots there actually are.

To find all of the possible rational real roots, take all possible factors of the constant term (+/-) and divide them by all possible factors of the leading coefficient. Then either substitute each of them in to the original function to see if you get "0", or use synthetic division to see if you get a remainder of "0". Unfortunately, none of the rational possibilities (including -1) works, so graphing the function will yield the best option to finding any of the real roots.

I hope this helps. Please let us know if you have further questions.

Answer 2

Up to 7 real roots.

Clearly x=-1 is a root. So (x+1) is factor. So do long division or synthetic division to find q(x) such that f(x) = (x+1)*q(x)