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Giải bài 35 trang 16 sách bài tập toán 10 - Cánh diều

Cho \({\left( {2x - \frac{1}{3}} \right)^4} = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + {a_4}{x^4}\). Tính:

Tổng hợp đề thi học kì 2 lớp 10𝕴 tất cả các mô🐻n - Cánh diều

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Đề bài

Cho \({\left( {2x - \frac{1}{3}} \right)^4} = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + {a_4}🐠{x^4}\). Tính: 

a) \({a_2}\)

b) \({a_0} + {a_1} + {a_2} + {a_3} + {a_4}\)

Phương pháp giải - Xem chi tiết

Bước 1: Áp dụng công thức khai triển: \({(a - b)^4} = {a^4} - 4{a^3}b + 6{a^2}{b^2} - 4a{b^3} + {b^4}\) với \(a = 2x,b = \frac{1}{3}\)

Bước 2: Thay x = 1 vào khai triển tꦗrong giả thiết để tính tổng các hệ số của kha♌i triển

Lời giải chi tiết

a) Ta có: 

 

                

Ta thấy \({a_2}\) là hệ số của \({x^2}\)Số hạng chứa \({x^2}\) trong khai triển biểu thức \({\left( {2x - \frac{1}{3}} \right)^4}\) là \(\frac{8}{3}{x^2}\)Suy ra hệ số của trong khai triển biểu thức \({\left( {2x - \frac{1}{3}} \right)^4}\) là \(\frac{8}{3}\)Vậy \({a_2} = \frac{8}{3}\)b) Ta có \({\left( {2x - \frac{1}{3}} \right)^4} = {a_0} + {a_1}x + {a_2}{x^2} + {a_3}{x^3} + {a_4}{x^4}\)

Chọn x = 1, ta được:

 

 

Vậy \({a_0} + {a_1} + {a_2} + {a_3} + {a_4} = \frac{{625}}{{81}}\)

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