To find the specified term in each binomial expansion, we use the binomial theorem. The general term T_r+1 in the expansion of (a+b)^n is given by:

To find the specified term in each binomial expansion, we use the binomial theorem. The general term T_r+1 in the expansion of (a+b)^n is given by: T_r+1 = nr a^n-r b^r a) Find the term x^2y^3 in (2x+y)^5. Step 1: Identify a, b, and n. In this expansion, a = 2x, b = y, and n = 5. We want the term with x^2y^3. Comparing b^r with y^3, we get r=3. Then n-r = 5-3 = 2. The term is T_3+1 = 53 (2x)^5-3 (y)^3. Step 2: Calculate the binomial coefficient and simplify the term. T_4 = 53 (2x)^2 y^3 53 = (5!)/(3!(5-3)!) = (5!)/(3!2!) = (5 × 4)/(2 × 1) = 10 (2x)^2 = 4x^2 T_4 = 10 · 4x^2 · y^3 = 40x^2y^3 The term is 40x^2y^3. b) Find the term x^2y^2 in (x-y)^4. Step 1: Identify a, b, and n. In this expansion, a = x, b = -y, and n = 4. We want the term with x^2y^2. Comparing b^r with (-y)^r and the desired y^2, we get r=2. Then n-r = 4-2 = 2. The term is T_2+1 = 42 (x)^4-2 (-y)^2. Step 2: Calculate the binomial coefficient and simplify the term. T_3 = 42 x^2 (-y)^2 42 = (4!)/(2!(4-2)!) = (4!)/(2!2!) = (4 × 3)/(2 × 1) = 6 (-y)^2 = y^2 T_3 = 6 · x^2 · y^2 = 6x^2y^2 The term is 6x^2y^2. c) Find the term xy^2 in (x-2y)^3. Step 1: Identify a, b, and n. In this expansion, a = x, b = -2y, and n = 3. We want the term with xy^2. Comparing b^r with (-2y)^r and the desired y^2, we get r=2. Then n-r = 3-2 = 1. The term is T_2+1 = 32 (x)^3-2 (-2y)^2. Step 2: Calculate the binomial coefficient and simplify the term. T_3 = 32 x^1 (-2y)^2 32 = (3!)/(2!(3-2)!) = (3!)/(2!1!) = (3)/(1) = 3 (-2y)^2 = (-2)^2 y^2 = 4y^2 T_3 = 3 · x · 4y^2 = 12xy^2 The term is 12xy^2. Send me the next one 📸