- #1
chaoseverlasting
- 1,050
- 3
Homework Statement
What is the coeff of [tex]x^{99}[/tex] in (x-1)(x-2)...(x-100)
2. The attempt at a solution
This has to do with the binomial coeff. I don't know how to go about it.
Werg22 said:Look at how a product develops as you add more terms,
if
[tex](x-1)(x-2)...(x-n) = x^n - (1+2+...+n)x^{n-1} + ... + (-1)^{n}*1*2*...*n [/tex]
then
[tex](x-1)(x-2)...(x-n)(x-(n+1)) = x^{n+1} - (1+2+...+n+n+1)x^{n} + ... + (-1)^{n+1}*1*2*...*n*(n+1) [/tex]
If we let n + 1 = m, then
[tex](x-1)(x-2)...(x-m) = x^m - (1+2+...+m)x^{m-1} + ... + (-1)^{m}*1*2*...*m [/tex]
The coefficient of a term is the numerical factor that is multiplied by a variable in an algebraic expression. It is typically represented by the letter "a" or "b" and is used to determine the amount or size of the term.
To calculate the coefficient of a term, you need to determine the number that is being multiplied by the variable. For example, in the term 3x, the coefficient is 3. In the term -5y, the coefficient is -5. If there is no number written explicitly, then the coefficient is assumed to be 1.
A coefficient is a number that is multiplied by a variable in an algebraic expression, while a constant is a number that stands alone in an expression and does not change. Coefficients can change depending on the variables in the expression, while constants remain the same.
The coefficient of the x term in a linear equation (y = mx + b) determines the slope of the line on a graph. A larger coefficient results in a steeper slope, while a smaller coefficient results in a flatter slope. The coefficient of the y term (b) determines the y-intercept, or where the line crosses the y-axis.
The coefficient of a term is important because it helps us understand the relationship between the variables in an algebraic expression. It allows us to determine the size or amount of a term, and also plays a crucial role in graphing linear equations and solving equations using algebraic methods.