rachelblanchette
contestada

Consider the following expression: (1 + x)^n. A) Use the Binomial Theorem to find the first four terms of this polynomial. B) Why is 1 + nx a good approximation of this expression when x is less than 1?

Respuesta :

Hagrid
Given:

The expression:  (1 + x)^n

The Binomial Theorem is used to predict the products of a binomial raised to a certain power, n, without multiplying the terms one by one. 

The following formula is used:

(a + b)^n = nCk * a^(n-k) * b^k

we have (1 +x)^n,

where a = 1 
            b = x
let n = 4

First term, k = 1

4C1 = 4

first term: 4*(1^(4-1))*x^1

Therefore, the first term is 4x. Do the same for the next three terms. 

2nd term: k =2 
3rd term: k = 3
4th term: k = 4

Answer:

part A answer below.. Sorry, I don't know the part B

Step-by-step explanation:

1st term  C(n,0) [tex]1^nx^0[/tex]  = [tex]\frac{n!}{(n-0)! (0!)}[/tex] [tex]1^nx^0[/tex] = 1 ·1 ·1 = 1

2nd term C(n,1) [tex]1^n^-^1x^1[/tex] = [tex]\frac{n!}{(n-1)!(1!)}[/tex] [tex]1^n^-^1x^1[/tex] = [tex]\frac{n!}{(n-1)!}x[/tex]

3rd term C(n,2) [tex]1^n^-^2x^2[/tex] = [tex]\frac{n!}{(n-2)!(2!)} 1^n^-^2x^2[/tex] = [tex]\frac{n!}{2(n-2)!}x^2[/tex]

4th term C(n,3)[tex]1^n^-^3x^3[/tex] = [tex]\frac{n!}{(n-3)!(3)!} 1^n^-^3x^3[/tex] = [tex]\frac{n!}{6(n-3)!} x^3[/tex]

Otras preguntas