Problem 34 - Algebra and Trigonometry

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x-2 y)^{9} $$

Step-by-Step Solution

Answer

The first three terms of the binomial expansion are \(x^9\), -18x^8y and 144x^7y^2.

Step 1: Define the coefficients, variables and powers

Identify the values of a, b and n in the problem. Here, a = x, b = -2y and n = 9. The coefficients for the first three terms can be found by using the binomial coefficient \[\binom{n}{k}\] which is equal to n!/(k!(n-k)!). Here, n is 9. For the first term k=0, second term k=1 and for the third term k=2.

Step 2: Calculate the first term

The first term is obtained when k=0 in the binomial coefficient, which simplifies to \[\binom{9}{0}\] and equals 1. Since a = x and n = 9 the first term becomes \[1*x^9*(-2y)^0\] which simplify to \(x^9\).

Step 3: Calculate the second term

The second term is obtained when k = 1. So, the binomial coefficient becomes \[\binom{9}{1}\] which equals 9. Thus the second term becomes \[9*x^8*(-2y)^1\] which simplifies to -18x^8y.

Step 4: Calculate the third term

The third term is obtained when k = 2. The binomial coefficient \[\binom{9}{2}\] equals 36. The third term becomes \[36*x^7*(-2y)^2\], which simplifies to 144x^7y^2.