SOLUTION: Let a, b, c, and dbe positive integers such that a < b < c < d.Suppose b = a + 2, c = a + 4, and d = a + 5. Write x^a + x^b + x^c + x^d in factored form.

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Question 513466: Let a, b, c, and dbe positive integers such that a < b < c < d.Suppose b = a + 2, c = a + 4, and d = a + 5. Write x^a + x^b + x^c + x^d in factored form.
Answer by drcole(72) (Show Source):
You can put this solution on YOUR website!
First, let's substitute in for b, c, and d, since we are given formulae for all three in terms of a:

Now let's remember how exponents add:

for all p and q. Thus we can rewrite our expression as:

Now let's look for ways to factor our expression. Notice that x^a is a common factor in each term, so we'll factor x^a out:

There doesn't appear to be any way to easily factor the remaining polynomial (the only possibilities for rational roots are 1 and -1, and neither of these work), so it looks like we're done.