補上出處
設\( f(x)=x^3+2x^2-3x-1 \),\( g(x)=x^4+3x^3-x^2-5x+2 \),且α、β、γ為\( f(x)=0 \)之三根。
(1)試求\( g(\alpha) \cdot g(\beta) \cdot g(\gamma) \)之值。
(2)試求\( \displaystyle \frac{1}{g(\alpha)}+\frac{1}{g(\beta)}+\frac{1}{g(\gamma)} \)之值。
(高雄女中雙週一題,96師大附中)
[提示]
(1)\( y=-x+3 \),用\( x=3-y \)代入得\( y^3-11y^2+36y-35=0 \)
三根之積\( g(\alpha) \cdot g(\beta) \cdot g(\gamma)=35 \)
(2)\( \displaystyle z=\frac{1}{y}=\frac{1}{-x+3} \)
取倒數得\( \displaystyle 1-11 \frac{1}{y}+36 \frac{1}{y^2}-35 \frac{1}{y^3}=0 \)
\( 1-11z+36z^2-35z^3=0 \)
三根之和\( \displaystyle \frac{1}{g(\alpha)}+\frac{1}{g(\beta)}+\frac{1}{g(\gamma)}=\frac{36}{35} \)
設\(f(x)=x^3+2x^2-3x-1\),\(g(x)=x^4+3x^3-x^2-5x+2\),且\(\alpha\)、\(\beta\)、\(\gamma\)為\(f(x)=0\)之三根,試求\(g(\alpha)\cdot g(\beta)\cdot g(\gamma)\)之值
。
(100全國高中聯招,
https://math.pro/db/thread-1163-1-1.html)
108.5.11補充
設\(f(x)=x^3-2x^2+3x-4\),\(g(x)=x^4-3x^3+5x^2-8x+1\),且\(\alpha\)、\(\beta\)、\(\gamma\)為\(f(x)=0\)之三根,試求\(g(\alpha)\cdot g(\beta)\cdot g(\gamma)\)之值
。
(108全國高中聯招,
https://math.pro/db/thread-3132-1-1.html)
109.6.2補充
設\(f(x)=x^3+2x^2-3x-1\),\(g(x)=x^4+3x^3-x^2-5x+1\),且\(\alpha\)、\(\beta\)、\(\gamma\)為\(f(x)=0\)之3根。試求:\(\displaystyle \frac{1}{g(\alpha)}+\frac{1}{g(\beta)}+\frac{1}{g(\gamma)}\)之值為
。
(109中壢高中代理,
https://math.pro/db/thread-3339-1-1.html)
111.5.14補充
設\(f(x)=x^3+3x^2-4x-2\),\(g(x)=x^4+6x^3+5x^2-16x-2\),且\(\alpha,\beta,\gamma\)為\(f(x)=0\)的三個根,則\(\displaystyle \frac{1}{g(\alpha)}+\frac{1}{g(\beta)}+\frac{1}{g(\gamma)}=\)
。
(111全國高中聯招,
https://math.pro/db/thread-3643-1-1.html)