Finding values with a polynomial function

Question

Knowing the following polynomial function:

y = 18x⁴ -81³ +109x² -26x -24

Find the value of x in which y is 1040

Answer 1

1. Plug in y = 1040 into the equation: 1040 = 18x⁴ - 81³ + 109x² - 26x - 24
2. Set the equation equal to zero by subtracting 1040 from both sides: 18x⁴ - 81³ + 109x² - 26x - 1064 = 0
3. Using the old fashioned way, at this point you have to start guessing at possible zeros of this equation. Typically you would use the rational roots theorem to find possible rational roots. Then use synthetic division to try them out and then learn what you can from each division. If you haven't learned the rational roots theorem or synthetic division, I'll leave it to you to look them up. Leave it to say that with graphing calculators, you can short circuit this trial and error method and plot the polynomial and look at possible zeros (or let the calculator solve for you. In this case the rational roots theorem says that there could be MANY so I'm going to just use the calculator. It says 4 is a zero. Lets use synthetic division with 4
4. The results show (x-4)(18x^3 - 9x^2 + 73x +266) are factors but the zeros of 18x^3 - 9x^2 + 73x +266 are irrational (approx -1.801653978432 and also two complex zeros.
5. So the best answer is 4 although the other 3 answers work as well