How to factorise this cubic

[James10492]

Hi I've got a cubic expression I need to factorise and I am really stuck. Currently I am doing graph transformations and the problem arose with a question where you are given a function f(x) = (x-1)(2x-3)(4-x) and you then have to give the function for the transformed graph f(x) + 2 in its fully factorized form.

The way I have approached this is to multiply out the brackets for the original function f(x), add 2, and then try to factorize the result (like breaking it apart and putting everything back together again). After multiplying out and adding 2 I get -2x³ +13x² -23x + 14, an expression where x isn't a factor. I have tried the trial and error method of finding a factor (by substituting numbers into f(x)) but to no avail. I can't help but feel that maybe I am trying too hard. Am I missing some kind of short cut method to factorise the new function with the transformation? This is a new topic for me so I wanted to make sure. Thanks for reading, any help would be greatly appreciated.

 

[Dr.Peterson]

Hi I've got a cubic expression I need to factorise and I am really stuck. Currently I am doing graph transformations and the problem arose with a question where you are given a function f(x) = (x-1)(2x-3)(4-x) and you then have to give the function for the transformed graph f(x) + 2 in its fully factorized form.

The way I have approached this is to multiply out the brackets for the original function f(x), add 2, and then try to factorize the result (like breaking it apart and putting everything back together again). After multiplying out and adding 2 I get -2x³ +13x² -23x + 14, an expression where x isn't a factor. I have tried the trial and error method of finding a factor (by substituting numbers into f(x)) but to no avail. I can't help but feel that maybe I am trying too hard. Am I missing some kind of short cut method to factorise the new function with the transformation? This is a new topic for me so I wanted to make sure. Thanks for reading, any help would be greatly appreciated.

The new function can't be factorized. Please quote the exact wording of the problem, because I don't think they can have said, "give the function for the transformed graph f(x) + 2 in its fully factorized form".

I will assume they told you to graph the new function by using the transformation, not by factoring it. So do that. That is the shortcut, as far as graphing is concerned!

Graph f(x) using the factorized form it is in, which is very helpful. Then think about how the new function g(x) = f(x) + 2 is related, and make that change to the graph you have drawn. The graph will convince you not to try factoring, if you ponder what you see.

An image of the actual problem will help us.