factor theorem

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harith

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Hi everyone.
I am self studying maths, from a book called 'Engineering Mathematics'. I am not that good in maths so I please bear with me if I ask stupid question. Here is my question: I need to find factors of 2x3 -x2 -16x +15. I have got two answers by trial and error method i.e (x+3)(x-1). The third answer is (2x -5), but i don't really know how to get (2x - 5). And I also divided (x+3) with 2x3-x2-16x+15 and got 2x2-7x+5. And I don't know how to factorize 2x2-7x+5. Please any help will be highly appreciated.
 
harith said:
Hi everyone.
I am self studying maths, from a book called 'Engineering Mathematics'. I am not that good in maths so I please bear with me if I ask stupid question. Here is my question: I need to find factors of 2x3 -x2 -16x +15. I have got two answers by trial and error method i.e (x+3)(x-1). The third answer is (2x -5), but i don't really know how to get (2x - 5). And I also divided (x+3) with 2x3-x2-16x+15 and got 2x2-7x+5. And I don't know how to factorize 2x2-7x+5. Please any help will be highly appreciated.
Click to expand...

You can divide you quadratic by (x-1).

I don't know what the "trial and error" method is but if you google for the rational roots theorem you can find all three roots just with a bit of arithmetic (assumig all roots are rational). Then you may transfer those roots into linear factors. For example, if you obtain x=A/B is a root, then AX-B will be a linear factor.

Another option would be to use the quadratic formula to obtain the roots of a quadratic, and translate that to a linear factor.
 
daon2 said:
You can divide you quadratic by (x-1).

I don't know what the "trial and error" method is but if you google for the rational roots theorem you can find all three roots just with a bit of arithmetic (assumig all roots are rational). Then you may transfer those roots into linear factors. For example, if you obtain x=A/B is a root, then AX-B will be a linear factor.

Another option would be to use the quadratic formula to obtain the roots of a quadratic, and translate that to a linear factor.
Click to expand...

The book assumes that I don't know the formula nor rational root. So i have to either solve by factoring or trial and error method. trial and error method is: lets say f(x) = 2x3-x2-16x+15, then we substitute x for number and see if the answer is zero. Thanks for the reply.
 
harith said:
The book assumes that I don't know the formula nor rational root. So i have to either solve by factoring or trial and error method. trial and error method is: lets say f(x) = 2x3-x2-16x+15, then we substitute x for number and see if the answer is zero. Thanks for the reply.
Click to expand...

One thing to note is that you can greatly reduce what numbers you guess as roots if you know ahead of time that some are likely to be integers.

f(x) = 2 x^3 - x^2 - 16 x + 15 = 0, so

2 x^3 - x^2 - 16 x = -15

x (2x^2 - x - 16) = -15

thus if x is a root it must divide -15, there are 8 integers that do this +/- 1,3,5,15. Plug these into f(x) and see if any result in 0.

once you find that root x0 just do the division f2(x) = f(x)/(x-x0)

Then repeat all this on f2(x) and keep repeating until either you've found all the factors or you've got a polynomial with non-integer roots.
 
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