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Giải bài 2 trang 39 Chuyên đề học tập Toán 10 – Chân trời sáng tạo

Tìm hệ số của \({x^{10}}\) trong khai triển của biểu thức \({(2 - x)^{12}}\)

Tổng hợp đề thi học kì 2 lớp ꧃10 tất cả các môn - Chân trời sánꦿg tạo

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Đề bài

Tìm hệ số của \({x^{10}}\) trong khai triển của biểu thức \({(2 - x)^{12}}\)

Phương pháp giải - Xem chi tiết

Công thức nhị thức Newton: \({(a + b)^n} = C_n^0{a^n} + C_n^1{a^{n - 1}}b + ... + C_n^{n - 1}a{b^{n - 1}} + C_n^n{b^n}\)

Lời giải chi tiết

Theo công thức nhị thức Newton, ta có:\(\begin{array}{l}{(2 - x)^{12}} = C_{12}^0{2^{12}} + C_{12}^1{2^{11}}\left( { - x} \right) + ... + C_{12}^k{2^{12 - k}}{\left( { - x} \right)^k} + ... + C_{12}^{12}{( - x)^{12}}\\ = C_{12}^0{2^{12}} - C_{12}^1{2^{11}}x + ... + {( - 1)^k}C_{12}^k{2^{12 - k}}{x^k} + ... + C_{12}^{12}{x^{12}}\end{array}\)Số hạng chứa \({x^{10}}\) ứng với \(k = 10\). Do đó hệ số của \({x^{10}}\)  là\(C_{12}^{10}{2^2}{( - 1)^{10}} = 66.4.1 = 264\)

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