Use the binomial theorem to find the term containing x^{12} in the expansion (-4 x-4)^{15}

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Step:1 ：Given the binomial expansion ((a+b)^n = sum_{k=0}^{n} {n choose k} a^{n-k} b^k), we are asked to find the term containing x^{12} in the expansion ((-4 x-4)^{15}). Here, a = -4x, b = -4, and n = 15.Step:2 ：We want to find the term where the power of x is 12, which means n-k = 12. Therefore, k = n - 12 = 15 - 12 = 3. So, we need to find the coefficient of the term where k = 3.Step:3 ：Substituting n = 15, k = 3, a = -4, and b = -4 into the binomial theorem, we find that the coefficient is -488552529920.Step:4 ：Thus, the term containing x^{12} in the expansion ((-4 x-4)^{15}) is (boxed{-488552529920x^{12}}).

Answer

Step: 1 ：Given the binomial expansion ((a+b)^n = sum_{k=0}^{n} {n choose k} a^{n-k} b^k), we are asked to find the term containing x^{12} in the expansion ((-4 x-4)^{15}). Here, a = -4x, b = -4, and n = 15.Step: 2 ：We want to find the term where the power of x is 12, which means n-k = 12. Therefore, k = n - 12 = 15 - 12 = 3. So, we need to find the coefficient of the term where k = 3.Step: 3 ：Substituting n = 15, k = 3, a = -4, and b = -4 into the binomial theorem, we find that the coefficient is -488552529920.Step: 4 ：Thus, the term containing x^{12} in the expansion ((-4 x-4)^{15}) is (boxed{-488552529920x^{12}}).

Use the binomial theorem to find the term containing $x^{12}$ in the expansion $(- 4 x - 4)^{15}$

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Use the binomial theorem to find the term containing x12 in the expansion (−4x−4)15

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Popular Problems

Use the binomial theorem to find the term containing $x^{12}$ in the expansion $(- 4 x - 4)^{15}$