Find the x^(16) th term (4x^(3)-3x^(2)+5x)^(7)

Question

Find the  x^(16) th term  (4x^(3)-3x^(2)+5x)^(7)

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Use Binomial Theorem: To find the x^(16) term in the expansion of (4x^(3)-3x^(2)+5x)^(7), we will use the binomial theorem, which states that the expansion of (a+b+c)^n can be found by considering the sum of terms of the form C(n, k1, k2, k3) * a^k1 * b^k2 * c^k3, where k1 + k2 + k3 = n and C(n, k1, k2, k3) is the multinomial coefficient. We need to find the combination of powers of 4x^3, -3x^2, and 5x that will give us x^16 when multiplied together.Consider General Term: First, let's consider the general term in the expansion of (a+b+c)^n, which is given by T(k1, k2, k3) = C(n, k1, k2, k3) * a^k1 * b^k2 * c^k3. In our case, a = 4x^3, b = -3x^2, and c = 5x. We need to find the values of k1, k2, and k3 such that 3k1 + 2k2 + k3 = 16 and k1 + k2 + k3 = 7.Find Largest Power: Let's start by finding the largest power of x that can be part of the x^16 term. Since the largest power of x in the binomial is x^3, we will first try k1 = 5, because 5 * 3 = 15, which is the closest to 16 without going over. This would leave us with x^1 to be obtained from the remaining terms.Calculate Remaining Terms: With k1 = 5, we have used up 5 of the 7 available terms, so we have k1 + k2 + k3 = 5 + k2 + k3 = 7. This simplifies to k2 + k3 = 2. Now we need to find values of k2 and k3 such that 2k2 + k3 = 1, because we already have 15 from the 5 * 3x^3 terms and we need a total power of 16.Determine Values of k2 and k3: The only way to get 2k2 + k3 = 1 with k2 + k3 = 2 is if k2 = 0 and k3 = 1. This gives us the term with powers of x that add up to 16: (4x^3)^5 * (-3x^2)^0 * (5x)^1.Calculate Multinomial Coefficient: Now we calculate the multinomial coefficient C(7, 5, 0, 1) which is the same as the binomial coefficient C(7, 5) because k2 = 0 does not contribute to the number of combinations. C(7, 5) = 7! / (5! * (7-5)!) = 7 * 6 / (2 * 1) = 21.Write Out the Term: We can now write out the term: T(5, 0, 1) = 21 * (4x^3)^5 * (-3x^2)^0 * (5x)^1 = 21 * 4^5 * x^(3*5) * 1 * 5x = 21 * 1024 * x^15 * 5x = 21 * 1024 * 5 * x^16.Calculate Coefficient: Multiplying the constants together gives us the coefficient of the x^16 term: 21 * 1024 * 5 = 107520.Final Result: Therefore, the x^16 term in the expansion of (4x^(3)-3x^(2)+5x)^(7) is 107520 * x^16.

Find the $x^{16}$ th term $(4x^{3}-3x^{2}+5x)^{7}$

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Find the $x^{16}$ th term $(4x^{3}-3x^{2}+5x)^{7}$