Question

(a) Find the first four terms, in ascending powers of x, of the binomial expansion of (2+px)^(9). Given that the coefficient of the x^(3) term in the expansion is -84. (b) (i) Find the value of p. (2) (ii) Find the numerical values for the coefficients of the x and x^(2) terms. (5)

Answer 1

The value of p is found to be -1 when solving for the coefficient of the x^3 term (-84) in the binomial expansion of (2+px)^9. Subsequently, the coefficients for the x and x^2 terms are 144 and 1008, respectively.

Step-by-step explanation:

To find the first four terms of the expansion of (2+px)^9, we use the binomial theorem:

(a + b)^n = a^n + n*a^(n-1)*b + n*(n-1)/2!*a^(n-2)*b^2 + n*(n-1)*(n-2)/3!*a^(n-3)*b^3 + ...

For the given expression, a=2, b=px, and n=9. The first four terms are:

1. 2^9
2. 9*2^8*(px)
3. 9*8/2!*2^7*(px)^2
4. 9*8*7/3!*2^6*(px)^3

Given that the coefficient of x^3 is -84, we can find p by solving

9*8*7/3!*2^6*p^3 = -84

From this, we calculate that p = -1. Now, to find the coefficients of x and x^2 terms, we substitute p back into the respective terms:

1. Coefficient of x: 9*2^8*p
2. Coefficient of x^2: 9*8/2!*2^7*p^2

These calculations yield the coefficients 144 for the x term and 1008 for the x^2 term.