Expand the given power by using Pascal's triangle. (2k - b) 5

32k^5 - 80k^4·b + 80k^3·b^2 - 40k^2·b^3 + 10k·b^4 - b^5

Step-by-step explanation:

The relevant row of Pascal's triangle is the one with 5 (the exponent) as the second number. That row is ...

1 5 10 10 5 1

These are the multipliers used for the terms of the expanded sum.

Each term of (a+b) ^n is of the form a^ (n-j) ·b^j for j ranging from 0 to n. It is multiplied by the corresponding term in the row of Pascal's triangle.

(2k - b) ^5 = 1 (2k) ^5 (-b) ^0 + 5 (2k) ^4 (-b) ^1 + 10 (2k) ^3 (-b) ^2 + 10 (2k) ^2 (-b) ^3 + 5 (2k) ^1 (-b) ^4 + 1 (2k) ^0 (-b) ^5

= 32k^5 - 80k^4·b + 80k^3·b^2 - 40k^2·b^3 + 10k·b^4 - b^5