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Rút gọn các biểu thức: P = (5x − 1) + 2(1 − 5x)(4 + 5x) + (5x + 4)^2
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Giải SBT Toán 8 Bài 3, 4, 5: Những hằng đẳng thức đáng nhớ
Bài 3.3 trang 8 SBT Toán 8 Tập 1:
a) P = (5x − 1) + 2(1 − 5x)(4 + 5x) + (5x + 4)2;
b) Q = (x – y)3 + (y + x) 3 + (y – x) 3 – 3xy(x + y).
Lời giải:
a) P = (5x − 1) + 2(1 − 5x)(4 + 5x) + (5x + 4)2
= 5x – 1 + (2 – 10x).(4 + 5x) + (5x + 4)2
= 5x – 1 + 2.( 4 + 5x) – 10x. (4 + 5x) + (5x)2 + 2. 5x. 4 + 42
= 5x – 1 + 8 + 10x – 40x – 50x2 + 25x2 + 40x + 16
= (– 50x2 + 25x2) + (5x + 10x – 40x + 40x) + (– 1 + 8 + 16)
= – 25x2 + 15x + 23
Vậy P = – 25x2 + 15x + 23.
b) Q = (x – y)3 + (y + x)3 + (y – x)3 – 3xy(x + y)
= x3 – 3x2y + 3xy2 – y3 + y3 + 3y2.x + 3yx2 + x3 + y3 – 3y2.x + 3yx2 – x3 – 3x2y – 3xy2
= x3 – 3x2y + 3xy2 – y3 + y3 + 3.xy2 + 3x2.y + x3 + y3 – 3x.y2 + 3x2.y – x3 – 3x2y – 3xy2
= (x3 + x3 – x3)+ (– 3x2y + 3x2y + 3x2y – 3x2y) + (3xy2 + 3xy2 – 3xy2 – 3xy2) + (–y3 + y3 + y3)
= x3 + 0x2y + 0.xy2 + y3
= x3 + y3
Vậy Q = x3 + y3.