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Ecrire le quotient de deux fonctions affines sous la forme de la somme d'un nombre réel et du quotient d'un nombre réel par une fonction affine en utilisant la méthode d'identification    ressource 2713

Soit f la fonction définie pour tout x ] - ; 1 2 [ SequenceForm ] DirectedInfinity -1 ; 12 [ ] 1 2 ; + [ SequenceForm ] 12 ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "\[Infinity]"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", FractionBox["1", "2"], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", DirectedInfinity[-1], ";", Rational[1, 2], "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", FractionBox["1", "2"], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", Rational[1, 2], ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] DirectedInfinity -1 ; 12 [ SequenceForm ] 12 ; + DirectedInfinity 1 [ par f ( x ) = - 2 ( 10 x - 9 ) 5 ( 2 x - 1 ) .
On admet qu'il existe deux réels a et b tels que, pour tout x ] - ; 1 2 [ SequenceForm ] DirectedInfinity -1 ; 12 [ ] 1 2 ; + [ SequenceForm ] 12 ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "\[Infinity]"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", FractionBox["1", "2"], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", DirectedInfinity[-1], ";", Rational[1, 2], "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", FractionBox["1", "2"], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", Rational[1, 2], ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] DirectedInfinity -1 ; 12 [ SequenceForm ] 12 ; + DirectedInfinity 1 [ , f ( x ) = a + b 5 ( 2 x - 1 ) .
Soit Q la fonction affine définie pour tout réel x par Q ( x ) = α 1 x + α 0 telle que, pour tout x ] - ; 1 2 [ SequenceForm ] DirectedInfinity -1 ; 12 [ ] 1 2 ; + [ SequenceForm ] 12 ; + DirectedInfinity 1 [ TagBox[RowBox[List[InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", RowBox[List["-", "\[Infinity]"]], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", FractionBox["1", "2"], "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", DirectedInfinity[-1], ";", Rational[1, 2], "["], Rule[Editable, False]], "\[Union]", InterpretationBox[RowBox[List["\"]\"", "\[InvisibleSpace]", FractionBox["1", "2"], "\[InvisibleSpace]", "\";\"", "\[InvisibleSpace]", "\"+\"", "\[InvisibleSpace]", "\[Infinity]", "\[InvisibleSpace]", "\"[\""]], SequenceForm["]", Rational[1, 2], ";", "+", DirectedInfinity[1], "["], Rule[Editable, False]]]], HoldForm] SequenceForm x HoldForm SequenceForm ] DirectedInfinity -1 ; 12 [ SequenceForm ] 12 ; + DirectedInfinity 1 [ , a + b 5 ( 2 x - 1 ) = Q ( x ) 5 ( 2 x - 1 ) .

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

α 1 = α 0 =