Using Pascal’s triangle, expand each of the following expressions in ascending powers of x: (i) (a+x)^3 (ii) (2+x)^6. (6 marks)DICT MOD 1 July 2020

(i) To expand (a+x)^3 using Pascal’s triangle, we start by writing the binomial and the exponent in the top row of the triangle:

1 3

We can then use the numbers in the triangle to fill in the rows below:

1 3 1 3

The expansion of (a+x)^3 is obtained by multiplying each term in the binomial by the corresponding coefficient in the triangle and then summing the terms:

(a+x)^3 = 1a^3 + 3a^2x + 3ax^2 + x^3

(ii) To expand (2+x)^6 using Pascal’s triangle, we start by writing the binomial and the exponent in the top row of the triangle:

1 6

We can then use the numbers in the triangle to fill in the rows below:

1 6 1 6 1 6 1 6

The expansion of (2+x)^6 is obtained by multiplying each term in the binomial by the corresponding coefficient in the triangle and then summing the terms:

(2+x)^6 = 12^6 + 62^5x + 152^4x^2 + 202^3x^3 + 152^2x^4 + 62*x^5 + x^6

= 64 + 384x + 960x^2 + 1280x^3 + 960x^4 + 384x^5 + x^6