Question

Find the smallest value of n that produces at least one x term with a positive exponent, in the expansion of 5x^-20(3x^4 - 9x^3)^n , where n ∈ Ζ+

Answer 1

A binomial is a polynomial with two terms. We're going to look at the Binomial Expansion Theorem, a shortcut method of raising a binomial to a power.

(x+y)0 = 1

(x+y)1 = x + y

(x+y)2 = x2 + 2xy + y2

(x+y)3 = x3 + 3x2y + 3xy2 + y3

(x+y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4

(x+y)5 = x5 + 5x4y + 10x3y2 +10x2y3 + 5xy4 + y5

There are several things that you hopefully have noticed after looking at the expansion

There are n+1 terms in the expansion of (x+y)n

The degree of each term is n

The powers on x begin with n and decrease to 0

The powers on y begin with 0 and increase to n

The coefficients are symmetric

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