Use the binomial theorem to find the binomial expansion of the given expression (2x-3y)^5

  32x⁵ -240x⁴y +720x³y² -1080x²y³ +810xy⁴ -243y⁵

Step-by-step explanation:

Powers of the first term of the binomial decrease as powers of the second term increase. The sum of their exponents is always 5. The coefficients come from the 6th row of Pascal's triangle (the one with 5 in the second position).

The coefficients are (1, 5, 10, 10, 5, 1), so the expansion is ...

  (a+b)⁵ = a⁵ + 5a⁴b +10a³b² +10a²b³ +5ab⁴ +b⁵

where a=2x and b=-3y

Putting those values into the above expansion, we get ...

  (2x -3y)⁵ = 32x⁵ -240x⁴y +720x³y² -1080x²y³ +810xy⁴ -243y⁵

It can be easier to work with the ratios of terms. This way, you're not trying to figure out unfamiliar (high) powers of numbers at each step. The first term is (2x)⁵ = 32x⁵. The next term will be b/a = -3y/(2x) times that, multiplied by the ratio of adjacent coefficients from Pascal's triangle:

  first term: 32x⁵

  second term: (5/1)·(-3y/(2x))·first term = -240x⁴y

  third term: (10/5)·(-3y/(2x))·second term = 720x³y²

  fourth term: (10/10)·(-3y/(2x))·third term = -1080x²y³

  fifth term: (5/10)·(-3y/(2x))·fourth term = 810xy⁴

  sixth term: (1/5)·(-3y/(2x))·fifth term = -243y⁵