Coefficient of x⁹ from (x-1)(x-2)(x-3)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)

To find the coefficient of \(x^9\) in the expansion of \((x-1)(x-2)(x-3)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)\), we need to look at the term containing \(x^9\) in the product. 

The general form of each polynomial term when expanded will be of the form \(x^n\) times a constant. We need to find the product of all terms that will result in \(x^9\), which means we are picking 8 linear terms (constants) from the given factors and one \(x\) from the remaining.

The polynomial can be represented as:
\[P(x) = (x-1)(x-2)(x-3)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)\]

Each term \(x^k\) where \(k = 0, 1, 2, \ldots, 9\) in the expansion of \(P(x)\) will be the sum of products of selecting \(k\) \(x\)'s from the 9 factors and \((9-k)\) constants from the remaining. For the \(x^9\) term, we have:
\[ x^9 \cdot (-1)^{0}\text{Sum of product of all combinations of 0 constants} \]

So, the coefficient of \(x^9\) will be given by the sum of products of choosing one constant from each pair in the 8 remaining pairs. But here we are selecting one \(x\) out of the total 9 terms, leaving 8 constants to be multiplied.

To simplify, the coefficient of \(x^9\) can be found as follows:
\[ \sum_{i=1}^{9} (-1)^{0} \cdot 1 = \sum_{i=1}^{9} 1 \]

Where \(i\) ranges over all constants from the factors.
So, we sum all combinations of 8 factors excluding one \(x\), leading to:
\[1 + 2 + 3 + 5 + 6 + 7 + 8 + 9 + 10\]

Summing these constants:
\[1 + 2 + 3 + 5 + 6 + 7 + 8 + 9 + 10 = 51\]

Therefore, the coefficient of \(x^9\) is \(51\).

Would you like further details or have any questions?

Here are five questions you might ask next:
1. How do you find the coefficient of \(x^8\) from the same polynomial?
2. Can you explain how to expand a polynomial like this in general?
3. How does the binomial theorem apply to polynomial expansions?
4. What are the other coefficients of the polynomial \((x-1)(x-2)(x-3)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10)\)?
5. Can you show the process of expanding a simpler polynomial like \((x-1)(x-2)(x-3)\)?

**Tip:** When dealing with polynomial expansions, breaking down the problem into smaller steps and considering the contribution of each term can simplify complex calculations.

Learn how to find the coefficient of x^9 in the expansion of the polynomial (x-1)(x-2)(x-3)(x-5)(x-6)(x-7)(x-8)(x-9)(x-10) with a detailed solution. This problem involves polynomial expansion and coefficient determination, suitable for students in Grades 9-12.

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