Bob Valdez
01/22/2024 · Senior High School

2) En utilisant la méthode des coefficients indéterminés trouver les constantes réels a ; b ; c telles que \( \frac{f(x)}{g(x)}=x+a+\frac{b}{x-1}+\frac{c}{(x-1)^{2}} \)

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Pour trouver les constantes \( a \), \( b \) et \( c \) dans l'équation \( \frac{f(x)}{g(x)} = x + a + \frac{b}{x - 1} + \frac{c}{(x - 1)^2} \), suivez ces étapes : 1. **Multiplier les deux côtés par \( (x - 1)^2 \) pour éliminer les dénominateurs :** \[ f(x) = (x + a)(x - 1)^2 + b(x - 1) + c \] 2. **Développer et regrouper les termes :** \[ f(x) = x^3 + (-2 + a)x^2 + (1 - 2a + b)x + (a - b + c) \] 3. **Comparer les coefficients de \( f(x) \) avec \( p x^3 + q x^2 + r x + s \) :** \[ \begin{cases} p = 1 \\ q = -2 + a \\ r = 1 - 2a + b \\ s = a - b + c \end{cases} \] 4. **Résoudre le système pour \( a \), \( b \) et \( c \) :** - \( a = q + 2 \) - \( b = r + 2a - 1 \) - \( c = s - a + b \) Ainsi, en fonction des coefficients de \( f(x) \), vous pouvez déterminer les valeurs de \( a \), \( b \) et \( c \).

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