How do I factorize (x-1)(x-2)(x-3)(x-4) -48

[eureka_beyond]

Homework Statement

how do I factorize (x-1)(x-2)(x-3)(x-4) -48 into (x²-5x+12)(x²-5x-2)

Homework Equations

no relevant equations

The Attempt at a Solution

I've found the value of (x-1)(x-4) and the value of (x-2)(x-3) first. which is (x²-5x+4) abd (x²-5x+6), I put them together and -48 at the end. The problem is, how do I put the -48 inside the brackets?

 

[CompuChip]

Homework Statement

how do I factorize (x-1)(x-2)(x-3)(x-4) -48 into (x²-5x+12)(x²-5x-2)

Why would you need to put the -48 inside the brackets?

You are not asked to do so, are you?

 

[eureka_beyond]

if I don't put the -48 inside, then the answer won't be (x2-5x+12)(x2-5x-2), which is the answer to the question. So yes, I do need to put the -48 inside.

 

[symbolipoint]

Completely multiply the binomials in your first member and simplify, putting this into general form. Look then at one of your degree-two factors from your second member. You should be able to divide your second member by either of these factors. If you find no remainder, then you know the quotient found is the other factor. Read and reread that process very carefully and then do it part by part.

Your entire first member is (x-1)(x-2)(x-3)(x-4) -48 and your entire second member is
(x²-5x+12)(x²-5x-2)

 

[eureka_beyond]

Look then at one of your degree-two factors from your second member. You should be able to divide your second member by either of these factors.

I don't get it =[, sorry but can you explain a bit more, I'm really bad at this topic. Thankyou so much !

 

[eumyang]

How was the question worded exactly? You said:
"how do I factorize (x-1)(x-2)(x-3)(x-4) -48 into (x²-5x+12)(x²-5x-2)"

But was the factorization (x²-5x+12)(x²-5x-2) given in the problem, or are we looking for this factorization?

symbolipoint is assuming that the factorization was given. So given this:
(x-1)(x-2)(x-3)(x-4) - 48 = (x²-5x+12)(x²-5x-2)
You completely expand the left hand side and write the resulting polynomial in standard form (decreasing powers of x). Then divide it by one of the factors on the right hand side using long division. If this is a true statement, then the quotient you get should be the other factor.

However, if we are NOT given this factorization, and that we are supposed to find it ourselves, then symbolipoint's suggestion won't work.

 

[symbolipoint]

If the actual quadratic factored form is not given, or at least if one of the factors is not given, then we may be unable to find a factorization for the first expression, (x-1)(x-2)(x-3)(x-4) -48 , since both the quadratic factors appear to not be factorable themselves into first degree binomials.

We would need to actually be given the factorization and then show that this factorization is correct.

 

[eureka_beyond]

I should have put it clearer in the first place. The actual form (x2-5x+12)(x2-5x-2)
is not given.