Problem 33 - Algebra and Trigonometry

Question

Write the first three terms in each binomial expansion, expressing the result in simplified form. $$ (x-2 y)^{10} $$

Step-by-Step Solution

Answer

The first three terms in the binomial expansion of \((x - 2y)^{10}\) are \(x^{10}, -20x^{9}y, 180x^{8}y^{2}\).

Step 1: Identify the components of the binomial

In the binomial expression \((x - 2y)^{10}\), we have \(a = x\), \(b = -2y\), and \(n = 10\).

Step 2: Calculation of the first term

The first term will be where \(k = 0\). Use the formula to find that the first term is: \(\binom{10}{0} \cdot x^{10-0} \cdot (-2y)^0 = 1 \cdot x^{10} \cdot 1 = x^{10}\).

Step 3: Calculation of the second term

The second term will be where \(k = 1\). Use the formula to find that the second term is: \(\binom{10}{1} \cdot x^{10-1} \cdot (-2y)^1 = 10 \cdot x^{9} \cdot (-2y) = -20x^{9}y\).

Step 4: Calculation of the third term

The third term will be where \(k = 2\). Use the formula to find that the third term is: \(\binom{10}{2} \cdot x^{10-2} \cdot (-2y)^2 = 45 \cdot x^{8} \cdot 4y^{2} = 180x^{8}y^{2}\).