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# Solving a trigonometric cubic. watch

1. How can I solve the following for without using trial and improvement to find a factor and then factorising the resulting quadratic? :
2. let tan(theta) = x?
3. (Original post by Jingers)
let tan(theta) = x?
Yes, but how can I find the factors in order to factorise this.
also please note, I am NOT allowed to use a calculator,
4. Anyone thought of a way to do it?
5. any divisor of the form (x-a) has a|11 (can you see why).

this narrows down what you have to guess considerably.
6. Change tan(theta) to x:

Now, since the constant is 11, there are only four possible factors, 1, -1, 11 or -11

EDIT: Are you sure you've written the question properly. Because you couldn't possibly do this in a non-calculator paper. The answers are
7. Let .

First things first, it's evident that the only integer factors of 11 ( 1 & 11) are not roots themselves. So...

The coefficient of isn't one, so let's divide through to make it easier to handle, albeit more clumsy-looking:

Now we're dealing with fractions. As usual, any roots will be found as a factor of the constant term. Let's try some:

To get , what are the combinations you would intuitively try? Remember we are looking for only one of the roots...

Spoiler:
Show
Try , the most 'obvious' choice of factor for the constant term.
Substituting into the expression gives:

Hey presto. You can now reduce it to a quadratic, and use the formula to factorise that.

That's how I'd approach it, anyway. Can't see an easier way at the moment.

What a horrible question.
8. (Original post by james.h)
Let .

First things first, it's evident that the only integer factors of 11 ( 1 & 11) are not roots themselves. So...

The coefficient of isn't one, so let's divide through to make it easier to handle, albeit more clumsy-looking:

Now we're dealing with fractions. As usual, any roots will be found as a factor of the constant term. Let's try some:

To get , what are the combinations you would intuitively try? Remember we are looking for only one of the roots...

Spoiler:
Show
Try , the most 'obvious' choice of factor for the constant term.
Substituting into the expression gives:

Hey presto. You can now reduce it to a quadratic, and use the formula to factorise that.

That's how I'd approach it, anyway. Can't see an easier way at the moment.

What a horrible question.
Wow thanks for the help.
9. (Original post by Farhan.Hanif93)
Wow thanks for the help.

Don't know whether that'd work in all situations, though, so don't count on it.
10. (Original post by innerhollow)
Change tan(theta) to x:

Now, since the constant is 11, there are only four possible factors, 1, -1, 11 or -11

EDIT: Are you sure you've written the question properly. Because you couldn't possibly do this in a non-calculator paper. The answers are
Haha it's part of a STEP question so I'm not surprised it *seems* impossible. But the fact of the matter is that after finding x=0.5, I can get a quadratic to factorise and the formula does that for me
11. (Original post by Farhan.Hanif93)
How can I solve the following for without using trial and improvement to find a factor and then factorising the resulting quadratic? :

This cubic rings a bell. Does it by any chance emerge from a STEP question where one of the roots is suggested by the earlier part of the question?
12. I think I have it. When asked as STEP I 2005 Q4, you have to guess the root by noticing that f(0) and f(1) have different signs.

But anyone who tackled STEP I 2001 Q4 was given a big hint in the earlier part of the question, which 2005 candidates might have remembered.
13. That would make a lot more sense.
14. (Original post by around)
any divisor of the form (x-a) has a|11 (can you see why).
This is somewhat incomplete: http://en.wikipedia.org/wiki/Rational_root_theorem

I'll also note that "can you see why" is perhaps optimistic - I wouldn't say it's obvious.

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