Question

How do you use Pascal's triangle to expand `(x ^ { 3} y - 1) ^ { 6}`?

Answer 1

Solution for this problem is below:

Explanation:

We can use Pascal's triangle to solve this problem.

The triangle values for the exponent 3 are:

`color(red)(1)color(white)(.........)color(red)(6)color(white)(.........)color(red)(15)color(white)(.........)color(red)(20)color(white)(.........)color(red)(15)color(white)(.........)color(red)(6)color(white)(.........)color(red)(1)`

Therefore `(color(blue)(x^3y) - color(green)(1))^6` can be expanded as:

`color(red)(1)(color(blue)(x^3y))^6(color(green)(-1))^0 + color(red)(6)(color(blue)(x^3y))^5(color(green)(-1))^1 + color(red)(15)(color(blue)(x^3y))^4(color(green)(-1))^2 + color(red)(20)(color(blue)(x^3y))^3(color(green)(-1))^3 + color(red)(15)(color(blue)(x^3y))^2(color(green)(-1))^4 + color(red)(6)(color(blue)(x^3y))^1(color(green)(-1))^5 + color(red)(1)(color(blue)(x^3y))^0(color(green)(-1))^6`

`color(red)(1)(color(blue)(x^18y^6))1 + color(red)(6)(color(blue)(x^15y^5))(color(green)(-1)) + color(red)(15)(color(blue)(x^12y^4))color(green)(1) + color(red)(20)(color(blue)(x^9y^3))(color(green)(-1)) + color(red)(15)(color(blue)(x^6y^2))color(green)(1) + color(red)(6)(color(blue)(x^3y))(color(green)(-1)) + (color(red)(1)) (color(blue)(1))(color(green)(1))`

`x^18y^6 - 6x^15y^5 + 15x^12y^4 - 20x^9y^3 + 15x^6y^2 - 6x^3y + 1`