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Giải bài 25 trang 46 sách bài tập toán 7 - Cánh diều

Cho đa thức \(F(x) = {x^7} - \frac{1}{2}{x^3} + x + 1\)

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Đề bài

Cho đa thức \(F(x) = {x^7} - \frac{1}{2}{x^3} + x + 1\)

a) Tìm đa thức Q(x) sao cho F(x) + Q(x) = \({x^5} - {x^3} + 2\)

b) Tim đa thức R(x) sao cho F(x) – R(x) = 2

Phương pháp giải - Xem chi tiết

Thực hiện phép trừ hai đa thức một biến theo quy tắc với \(Q(x) = F(x) - ({x^5} - {x^3} + 2)\) và \(R(x) = F(x) - 2\)

Lời giải chi tiết

a) F(x) + Q(x꧅) = \({x^5} - {x^3} + 2\) \( \Rightarrow Q(x) = ({x^5} - {x^3} + 2) - F(x)\)

                                                   \( = ({x^5} - {x^3} + 2) - \left( {{x^7} - \frac{1}{2}{x^3} + x + 1} \right) = {x^5} - {x^3} + 2 - {x^7} + \frac{1}{2}{x^3} - x - 1\)                                                   \( =  - {x^7} + {x^5} + \left( {\frac{1}{2} - 1} \right){x^3} - x + (2 - 1) =  - {x^7} + {x^5} - \frac{1}{2}{x^3} - x + 1\)Vậy \(Q(x) =  - {x^7} + {x^5} - \frac{1}{2}{x^3} - x + 1\)

b) F(x) – R(xꩵ) = 2 \( \Rightarrow R(x) = F(x) - 2 = \left( {{x^7} - \frac{1}{2}{x^3} + x + 1} \right) - 2\)\( = {x^7} - \frac{1}{2}{x^3} + x + 1 - 2 = {x^7} - \frac{1}{2}{x^3} + x - 1\)

Vậy \(R(x) = {x^7} - \frac{1}{2}{x^3} + x - 1\)

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