How Do I Solve Polynomial Equations with Real and Complex Roots?

[crookesm]

Given:

$$s(t) = 1/2t^3-5t^2+3t+6$$

I'm trying to find all values of t where s(t) = -30

My first thought is to solve for 0 hence:

$$1/2t^3-5t^2+3t+36=0$$

I know the answers are t=4 and t=8.196 but I can't get to it...I'm assuming I need to factor this down but I'm can't see it. Any help/hints would be most appreciated as I've been banging my head against a brick wall for some time now...

 

[arildno]

You know that 4 is a root of the polynomial in your last equation.
Use polynomial division to compete the factoring.

 

[chris_tams]

As above, but divide (x-4) into 1/2t^3-5t^2+3t+36

Then when you get the second polynomial use the quadratic solution to get t=8.196

 

[crookesm]

Thanks for the hints - reading up on polynomial division (which I wasn't familiar with) I have found that the factors are:

$$(t-4)(1/2t^2-3t-9)$$

However, (and I realize I'm getting slightly off topic here) how would I even arrive at (t-4) being one of the original factors. Using GCF I can easily see that:

$$(t)(1/2t^2-5t+3)+36 = 0$$

But the jump to t-4 has got me stumped!

Any pointers to threads/websites on factoring polynomials/finding roots of polynomials would be most appreciated.

 

[arildno]

I believe Fermat developed various techniques in order to make good, reasoned guesses for the roots of polynomials.

However, when meeting a polynomial in a textbook that you don't immediately recognize the roots of, remember that the constant term is the product of the roots.

Therefore, one way of arriving at 4 as a root is to examine the integer factors of 36, and see which of these (if any!) might be a root.

 

[jatin9_99]

putting the equation equal to -30 u can make a cubic equation which will be then easy to solve for

 

[mccp]

Use descarte's rule of signs to find the nature of the roots (positive, negative, or complex).

After that take X minus all factors (positive and negative) of the coefficient of the highest degree of your variable over the factors of your constant.

3X^3 + 5X^2 - 2X + 8 =0

so your possible (real) factors are going to be (3/8), (3/4), (3/2), 3, (1/8), (1/4), (1/2), 1 and their inverses.

Well, now you're left with 14 possible roots, you can a) try them all algorithmically, or b) apply some heuristics. Graph the polynomial (preferably on a calculator) and look at which of the roots seem to be true (where the funtion intersects the X-Axis)--After that, try each of your roots until you find the root you're looking for.

Divide the equation by your new-found root and find the other two from your quadratic.

Hope I helped.