Question

I'm using the rational zeros theorem to find all the real zeros of this function: f(x)=11x^4+10x^3-34x^2-30x+3. I did the whole p/q thing and got +-1, +-3, +-1/11, and +-3/11. Only 1/11 worked but apparently I haven't found all the real zeros. What am I missing?

Answer 1

Did you divide out that (x-r) and repeat the process?

Answer 2

x-r?

Answer 3

If you have a polynomial with roots (zeros) \(r_1,r_2, ... r_n\) it can be written as \[P(x) = (x-r_1)(x-r_2)...(x-r_n)\]and so after you find one of the zeros or roots, you can divide by that product term and repeat the process on a simpler polynomial.

Answer 4

The highest power term in this polynomial is 4, so there will be a total of 4 zeros. We can see how many of them should be real with Descartes' Rule of Signs.

Write the polynomial in descending order (highest power to lowest power)
count the number of times the sign changes scanning from left to right
That's is the number of positive real roots, though it may be decreased by a multiple of 2.

Here we have ++--+ for the signs, so there are 2 sign changes. That means we have 2 positive, real roots, or 0 positive, real roots.

Next, rewrite the polynomial as P(-x). Essentially, you just flip the sign on the odd-power terms:
+--++
Again, two sign changes, so we have either 2 negative, real roots, or 0 negative, real roots, because we may have some number of pairs of complex roots.

Finally, the total root count has to equal the highest exponent, with the remainder not already counted being made up by complex roots. That means our possible scenarios are:

2 positive real, 2 negative real, 0 complex
0 positive real, 2 negative real, 2 complex
2 positive real, 0 negative real, 2 complex
0 positive real, 0 negative real, 4 complex

except you've already found a positive real root, so the 2nd and 4th possibilities are ruled out.

Answer 5

Did you try -1?

Answer 6

Yeah, I was looking at my work and saw that I made a mistake. -1 is a zero.

Answer 7

there's a clever method for evaluating polynomials that helps considerably when doing this sort of thing (assuming you don't have a computer to do it for you)

interested in hearing about it?

Answer 8

Write out your polynomial in descending order, but make sure that any missing terms are written out as well, with 0 as the coefficient. So \(x^2-4\) would be written \(x^2+0x-4\)

Now, take your value of x that you want to evaluate. Multiply it by the first coefficient. Add the next coefficient. Multiply by x again. Add the next coefficient. Multiply by x again. Keep going until you've added the constant term. The result is the value of the polynomial at x, but with many fewer steps. Evaluating your polynomial at x = 2, for example, I would think "11*2 = 22, + 10 = 32 , *2 = 64 , - 34 = 30, *2 = 60, - 30 = 30, *2 = 60, + 3 = 63."

And if we do it the hard way, \[11(2)^4+10(2)^3-34(2)^2-30(2)+3 \]\[= 11*16+10*8-34*4-30*2+3 = 176+80-136-60+3 = 63\]