Factoring problem that has me stumped

  • Thread starter crookesm
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    Factoring
In summary, the author is trying to find all values of t where s(t)=-30 and has found that t=4 and t=8.196. However, he is stumped as to how to find t-4.
  • #1
crookesm
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Given:

[tex] s(t) = 1/2t^3-5t^2+3t+6 [/tex]

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

My first thought is to solve for 0 hence:

[tex] 1/2t^3-5t^2+3t+36=0 [/tex]

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...
 
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  • #2
You know that 4 is a root of the polynomial in your last equation.
Use polynomial division to compete the factoring.
 
  • #3
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
 
  • #4
Thanks for the hints - reading up on polynomial division (which I wasn't familiar with) I have found that the factors are:

[tex] (t-4)(1/2t^2-3t-9) [/tex]

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:

[tex] (t)(1/2t^2-5t+3)+36 = 0 [/tex]

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.
 
Last edited:
  • #5
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.
 
  • #6
putting the equation equal to -30 u can make a cubic equation which will be then easy to solve for
 
  • #7
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.
 

1. What is factoring and why is it important?

Factoring is the process of breaking down a number or algebraic expression into its smaller factors. It is important because it helps us simplify complex expressions, solve equations and identify patterns in numbers.

2. How do I know when to use factoring to solve a problem?

If you are given a polynomial expression or an equation with one or more variables, it is likely that factoring will be necessary to solve it. Look for common factors or patterns in the expression to determine if factoring is the best approach.

3. What are the different methods of factoring?

There are several methods of factoring, including greatest common factor (GCF), difference of squares, trinomial factoring, and grouping. The method you use depends on the type of expression or equation you are trying to factor.

4. What should I do if I am stuck on a factoring problem?

If you are having trouble factoring a problem, try breaking it down into smaller parts and looking for common factors. You can also try using a different factoring method or asking a classmate or teacher for help.

5. Can factoring be used in real-life situations?

Yes, factoring can be used in various real-life situations, such as calculating interest rates, solving problems in finance and economics, and finding the dimensions of objects in architecture and engineering. It is a useful skill to have in many fields of study and professions.

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