5.5-bo'lim : qisman kasrlar
3. Quyidagi ifodalarning har birining qisman kasr parchalanishini aniqlang.
\[\frac{{125 + 4x - 9{x^2}}}{{\left( {x - 1} \right)\left( {x + 3} \right)\left( {x + 4} \right)}}\]Yechim
Birinchi qadam qisman fraksiyaning parchalanish shaklini aniqlashdir. Ushbu muammo uchun qisman kasr parchalanishi,
\[\frac{{125 + 4x - 9{x^2}}}{{\left( {x - 1} \right)\left( {x + 3} \right)\left( {x + 4} \right)}} = \frac{A}{{x - 1}} + \frac{B}{{x + 3}} + \frac{C}{{x + 4}}\]2-qadam
Ushbu ifoda uchun LCD displey \(\left( {x - 1} \right)\left( {x + 3} \right)\left( {x + 4} \right)\). Shartlarni qo'shish zaxira nusxasini beradi,
\[\frac{{125 + 4x - 9{x^2}}}{{\left( {x - 1} \right)\left( {x + 3} \right)\left( {x + 4} \right)}} = \frac{{A\left( {x + 3} \right)\left( {x + 4} \right) + B\left( {x - 1} \right)\left( {x + 4} \right) + C\left( {x - 1} \right)\left( {x + 3} \right)}}{{\left( {x - 1} \right)\left( {x + 3} \right)\left( {x + 4} \right)}}\]3-qadam
Numeratorlarni teng belgilash,
\[125 + 4x - 9{x^2} = A\left( {x + 3} \right)\left( {x + 4} \right) + B\left( {x - 1} \right)\left( {x + 4} \right) + C\left( {x - 1} \right)\left( {x + 3} \right)\]4-qadam
Endi biz konstantalarni aniqlash uchun \(x\) ning "yaxshi" qiymatlarini tanlashimiz kerak. Mana o'sha ish.
\[\begin{array}{l}{x = - 4:}\\{x = - 3:}\\{x = 1:}\end{array}\hspace{0.25in}\begin{aligned} - 35 & = 5C\\32 &= - 4B\\120 & = 20A\end{aligned}\hspace{0.25in} \to \hspace{0.25in}\begin{array}{l}{A = 6}\\{B = - 8}\\{C = - 7}\end{array}\]5-qadam
Bunda qisman fraksiya parchalanishi,
\[\require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{125 + 4x - 9{x^2}}}{{\left( {x - 1} \right)\left( {x + 3} \right)\left( {x + 4} \right)}} = \frac{6}{{x - 1}} - \frac{8}{{x + 3}} - \frac{7}{{x + 4}}}}\]