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Ecrire le quotient d'une fonction trinôme du second degré par une fonction affine sous la forme de la somme d'une fonction affine et du quotient d'un nombre réel par une fonction affine en utilisant la méthode d'identification    ressource 2714

Soit f la fonction définie pour tout x ] - ; - 9 10 [ SequenceForm ] DirectedInfinity -1 ; -910 [ ] - 9 10 ; + [ SequenceForm ] -910 ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "\[Infinity]"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", RowBox[List["-", FractionBox["9", "10"]]], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", DirectedInfinity[-1], ";", Rational[-9, 10], "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", FractionBox["9", "10"]]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", Rational[-9, 10], ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] DirectedInfinity -1 ; -910 [ SequenceForm ] -910 ; + DirectedInfinity 1 [ par f ( x ) = 30 x 2 + 37 x + 1 10 x + 9 .
On admet qu'il existe trois réels a, b et c tels que, pour tout x ] - ; - 9 10 [ SequenceForm ] DirectedInfinity -1 ; -910 [ ] - 9 10 ; + [ SequenceForm ] -910 ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "\[Infinity]"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", RowBox[List["-", FractionBox["9", "10"]]], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", DirectedInfinity[-1], ";", Rational[-9, 10], "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", FractionBox["9", "10"]]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", Rational[-9, 10], ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] DirectedInfinity -1 ; -910 [ SequenceForm ] -910 ; + DirectedInfinity 1 [ , f ( x ) = b + a x + c 10 x + 9 .
Soit Q la fonction trinôme du second degré définie pour tout réel x par Q ( x ) = α 2 x 2 + α 1 x + α 0 telle que, pour tout x ] - ; - 9 10 [ SequenceForm ] DirectedInfinity -1 ; -910 [ ] - 9 10 ; + [ SequenceForm ] -910 ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "\[Infinity]"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", RowBox[List["-", FractionBox["9", "10"]]], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", DirectedInfinity[-1], ";", Rational[-9, 10], "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", FractionBox["9", "10"]]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", Rational[-9, 10], ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] DirectedInfinity -1 ; -910 [ SequenceForm ] -910 ; + DirectedInfinity 1 [ , b + a x + c 10 x + 9 = Q ( x ) 10 x + 9 .

Exprimez les coefficients α 2 , α 1 , α 0 de Q en fonction de a, b et c.

α 2 = α 1 = α 0 =