\(\left(x-1\right)\left(x-3\right)\left(x-5\right)\left(x-7\right)-20\)
\(=\left[\left(x-1\right)\left(x-7\right)\right]\left[\left(x-3\right)\left(x-5\right)\right]-20\)
\(=\left(x^2-8x+7\right)\left(x^2-8x+15\right)-20\)
Đặt \(x^2-8x+7=t\),ta có :
\(t\left(t+8\right)-20\)
\(=t^2+8t-20\)
\(=\left(t^2+8t+16\right)-16-20\)
\(=\left(t+4\right)^2-36\)
\(=\left(t+4\right)^2-6^2\)
\(=\left(t+4-6\right)\left(t+4+6\right)\)
\(=\left(t-2\right)\left(t+10\right)\)
\(=\left(x^2-8x+7-2\right)\left(x^2-8x+7+10\right)\)
\(=\left(x^2-8x+5\right)\left(x^2-8x+17\right)\)
(x - 1)(x - 3)(x - 5)(x - 7) - 20
= [(x - 1)(x - 7)][(x - 3)(x - 5)] - 20
= (x2 - 7x - x + 7)(x2 - 5x - 3x + 15) - 20
= (x2 - 8x + 7)(x2 - 8x + 15) - 20
Đặt x2 - 8x + 7 = y, ta có:
y(y + 8) - 20
= y2 + 8y - 20
= y2 - 2y + 10y - 20
= (y2 - 2y) + (10y - 20)
= y(y - 2) + 10(y - 2)
= (y + 10)(y - 2)
= (x2 - 8x + 7 + 10)(x2 - 8x + 7 - 2)
= (x2 - 8x + 17)(x2 - 8x + 5)
