Board 8 > Board 8 math people, I need help.

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I'm taking a test tomorrow outside of my main class that covers material I either haven't learned yet or forgot about. I have the explanation for how to solve it, but there's a detail I don't understand.

Question:
Solve the following problem by recognizing the binomial expansion: X^7 + 28X^6 + 336X^5 + 2240X^4 + 8960X^3 + 21504X^2 + 28672X + 16374 = 0

Explanation:
Since the equation is of degree 7, we can use the row of Pascal's triangle that involves expansion to the seventh power. The coefficients are 1, 7, 21, 35, 21, 7, 1.

Dividing the coefficient of the second term (28) by 7 yields 4. Similarly, dividing the third coefficient (336) by 21 produces 16, which is 4 squared. Therefore, the problem can be reduced to (X + 4)^7 = 0. The solution is X = -4.

I don't really understand how dividing the coefficients of the second and third terms by their corresponding squares in Pascal's Triangle gets you (x+4)^7. I see the x+4 part, which is 28/7, but I don't see where you're getting the (x+4)^7 or where the 4^2=16 is coming into play. I also don't get why we're apparently ignoring the rest of the polynomial.
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Since they're telling you it's a binomial expansion you know it should factor as (x+a)^7.

You could pick any term to get a:
(x+a)^7 = x^7 + (7/1)*a*x^6+((7*6)/(2*1))*a^2*x^5+((7*6*5)/(3*2*1))*a^3*x^4 + ... + a^7, so really you would have to check that all six of 7*a=28, 21*a^2=336, ..., a^4=16374 give you the same solution for a. Because they've told you that it _is_ a binomial expansion, you can pick any one of them to solve for a.
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Ahh, thank you.

I don't think I learned about Pascal's Triangle in Algebra (it wasn't familiar) and I've been hitting youtube videos to learn about it.
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