91Ó°ÊÓ

Understanding coefficients is key to solving polynomial-related problems. In a polynomial expansion, the coefficient is the numerical factor that multiplies the variable part of each term.
For example, in the term \(5x^{2}\), 5 is the coefficient, and \(x^{2}\) is the variable part.

• Coefficients help determine the value of a term.
• When multiple terms are combined, each coefficient plays a role in forming the overall polynomial.
  

In the problem given, we need to find the coefficient of \(a^3 b^7\). Identifying this coefficient involves understanding the numerical part that will multiply \(a^3 b^7\) after expanding \((a+b+2c)^{10}\). Using the multinomial theorem, we isolate this term and calculate the coefficient by considering the factorials, which leads us to find that the coefficient is 120.

Polynomial expansion is the process of expressing a polynomial raised to a power as a sum of terms. Each term consists of variables raised to powers, multiplied by coefficients.
The multinomial theorem is a generalization of the binomial theorem and is used to expand polynomials with more than two terms. It states that:
\begin{aligned}(x_1 & +x_2+...+x_m)^{n} = \ & \sum_{k_1+k_2+...+k_m=n} \frac{n!}{k_1!k_2!...k_m!}x_1^{k_1} x_2^{k_2}...x_m^{k_m}umberthis\text{}\rightØ\ \right), where \(k_1+k_2+...+k_m=n\).
In our specific problem, we expand \((a+b+2c)^{10}\). We are interested in the term involving \(a^3 b^7\). With the general formula in mind:

• Identify the powers: \(a^3\), \(b^7\), and \((2c)^0\)
• Apply the theorem: Use \(k_1=3\), \(k_2=7\), \(k_3=0\) for simplicity.

Then, focus on the coefficient calculation, which tells us how these terms combine numerically.

Factorials are crucial in combinatorics and polynomial expansions. They are used to compute coefficients in polynomial terms using formulas like those in the multinomial theorem.
A factorial, denoted as \(n!\), is the product of all positive integers up to \(n\). For instance, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\).

• Key properties: \(0! = 1\) by definition.
• Factorials grow very quickly with larger numbers.
  

In the context of our problem, factorials are used to find the coefficient of specific terms. We calculate \( \frac{10!}{3!7!}\), which involves:

• Calculating \(10! = 3628800\)
• Calculating \(3! = 6\)
• Calculating \(7! = 5040\)

Then, combining these: \( \frac{3628800}{6 \times 5040} = 120\). This step shows how factorials are instrumental in simplifying the coefficients during polynomial expansion.