The coefficient of x cube in the expansion of (3+az)6 is 160 find the value of the constant a, hence find the coefficient of x cube in the expansion of (3+ ax)6(1-2x)

Here's a step-by-step solution to the problem. I will assume that 'z' in the first part of the question is a typo and should be 'x', consistent with the second part of the question. Part 1: Find the value of the constant a. We are given that the coefficient of x^3 in the expansion of (3+ax)^6 is 160. Step 1: Write the general term of the binomial expansion. The general term in the expansion of (A+B)^n is given by T_r+1 = nr A^n-r B^r. For (3+ax)^6, we have A=3, B=ax, and n=6. So, the general term is: T_r+1 = 6r (3)^6-r (ax)^r T_r+1 = 6r 3^6-r a^r x^r Step 2: Find the term containing x^3. For the term with x^3, we set r=3: T_3+1 = T_4 = 63 3^6-3 a^3 x^3 T_4 = 63 3^3 a^3 x^3 Step 3: Calculate the binomial coefficient and simplify. 63 = (6!)/(3!(6-3)!) = (6 × 5 × 4)/(3 × 2 × 1) = 20 3^3 = 27 Substitute these values back into the expression for T_4: T_4 = 20 × 27 × a^3 x^3 T_4 = 540 a^3 x^3 The coefficient of x^3 is 540a^3. Step 4: Solve for a. We are given that the coefficient of x^3 is 160: 540 a^3 = 160 a^3 = (160)/(540) a^3 = (16)/(54) a^3 = (8)/(27) Take the cube root of both sides: a = 3/(27) a = (2)/(3) The value of the constant a is (2)/(3). Part 2: Find the coefficient of x^3 in the expansion of (3+ax)^6(1-2x). Now we use the value a=(2)/(3). The expression becomes (3+(2)/(3)x)^6(1-2x). Step 1: Expand (3+(2)/(3)x)^6 up to the x^3 term. We need the coefficients of x^0, x^1, x^2, and x^3 from (3+(2)/(3)x)^6. Using the general term T_r+1 = 6r 3^6-r ((2)/(3)x)^r = 6r 3^6-r ((2)/(3))^r x^r: Coefficient of x^0 (constant term, r=0): 60 3^6 ((2)/(3))^0 = 1 × 729 × 1 = 729 Coefficient of x^1 (r=1): 61 3^5 ((2)/(3))^1 = 6 × 243 × (2)/(3) = 6 × 81 × 2 = 972 Coefficient of x^2 (r=2): 62 3^4 ((2)/(3))^2 = 15 × 81 × (4)/(9) = 15 × 9 × 4 = 540 Coefficient of x^3 (r=3): 63 3^3 ((2)/(3))^3 = 20 × 27 × (8)/(27) = 20 × 8 = 160 So, (3+(2)/(3)x)^6 = 729 + 972x + 540x^2 + 160x^3 + Step 2: Multiply the expansion by (1-2x) and identify x^3 terms. We need the coefficient of x^3 in (729 + 972x + 540x^2 + 160x^3 + )(1-2x). To get an x^3 term, we can multiply: The x^3 term from the first bracket by the constant term from the second bracket: (160x^3) × (1) = 160x^3 The x^2 term from the first bracket by the x term from the second bracket: (540x^2) × (-2x) = -1080x^3 Step 3: Sum the coefficients of the x^3 terms. The total coefficient of x^3 is the sum of these contributions: 160 + (-1080) = 160 - 1080 = -920 The coefficient of x^3 in the