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Giải bài 36 trang 50 sách bài tập toán 7 - Cánh diều

Xét đa thức \(P(x) = (2{x^2} + a)(2{x^3} - 3) - 5a(x + 3) + 1\) (với a là một số).

ౠTổng hợp đề thi học kì 2 lớp 7 tất cả các môn - Cánh diều

Toán - Văn - Anh - Khoa học tự nhiên...
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Đề bài

Xét đa thức \(P(x) = (2{x^2} + a)(2{x^3} - 3) - 5a(x + 3) + 1\) (với a là một số).

a) Thu gọn và sắp xếp đa thức P(x) theo số mũ giảm dần của biến

b) Tìm a sao cho tổng các hệ số của đa thức P(x) bằng – 37

Phương pháp giải - Xem chi tiết

Bước 1: Rút gọn đa thức P(x)

Bước 2: Tìm tổng các hệ số của đa thức P(x) là một đa thức Q(a) với biến a

Bước 3: Tìm a sao cho Q(a) + 37 = 0

Lời giải chi tiết

a) \(P(x) = (2{x^2} + a)(2{x^3} - 3) - 5a(x + 3) + 1\)\( = 4{x^5} - 6{x^2} + 2a{x^3} - 3a - 5ax - 15a + 1\)                                                          \( = 4{x^5} + 2a{x^3} - 6{x^2} - 5ax + 1 - 18a\)

b) Tổng các hệ số của đa thức P(x) là: \(4 + 2a - 6 - 5a + 1 - 18a =  - 21a - 1\)

Theo giả thiết, \( - 21a - 1 =  - 37 \Rightarrow  - 21a =  - 37 + 1 \Rightarrow  - 21a =  - 36 \Rightarrow a = \frac{{12}}{7}\)

Vậy với \(a = \frac{{12}}{7}\) thì tổng các hệ số của đa thức P(x) bằng – 37

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