How to factor this monster into two quadratics?

[motornoob101]

Homework Statement

How to factor this monster? I am rather clueless
$$144t^{4}-288t^{3}+864t^{2}-720t+900$$

Homework Equations

Hint is given that you can get two quadratic polynomials out of it..

The Attempt at a Solution

I can always cheat by using Maple to factor but I don't want do it! I want learn how to do it.

Thanks for any help!

 

[tiny-tim]

Hi motornoob!

Well, for a start, you can divide by 36 to get:

$$4t^{4}\,-\,8t^{3}\,+\,24t^{2}\,-\,20t\,+\,25$$​

hmm … now to think …

 

[motornoob101]

Yeah I divided by 144 to get something smaller but the last number is like 6.25.. I said is no big deal.. but still, I can't factor this.

 

[tiny-tim]

Yeah I divided by 144 to get something smaller but the last number is like 6.25.. I said is no big deal.. but still, I can't factor this.

motornoob, there's no point in making fractions - the object is to simplify it!

Stick with mine:

$$4t^{4}\,-\,8t^{3}\,+\,24t^{2}\,-\,20t\,+\,25$$​

I think the best way to do this is just to write it as

$$(at^{2}\,+\,bt\,+\,c)(dt^{2}\,+\,et\,+\,f)\,,$$​

and just multiply it out, giving you five equations to solve.

(I've done it - you can more-or-less guess the right answer once you've done that.)

Have a go!

 

[motornoob101]

Five equations to solve huh. I sure hope they don't give us something like this on an exam, considering this part of an integration problem. Thanks so much for the help.