Exercice 2-1

Calculer les sommes algébriques des fractions suivantes :

a)  ${\displaystyle {\frac {4ab+2b^{2}-12a^{2}}{3(a^{2}-b^{2})}}+{\frac {2a-b}{a-b}}+{\frac {7a}{3(a+b)}}}$

b)  ${\displaystyle {\frac {a+b}{2(a-b)}}-{\frac {a-b}{2(a+b)}}+{\frac {2b^{2}}{a^{2}-b^{2}}}}$

c)  ${\displaystyle {\frac {1}{a^{2}-3b^{2}+2ab}}+{\frac {1}{b^{2}-3a^{2}+2ab}}-{\frac {2}{3a^{2}+10ab+3b^{2}}}}$

d)  ${\displaystyle {\frac {2(7x-4)}{6x^{2}-7x+2}}+{\frac {x-10}{6x^{2}-x-2}}-{\frac {2(4x-1)}{4x^{2}-1}}}$

e)  ${\displaystyle {\frac {a+b}{ax+by}}+{\frac {a-b}{ax-by}}+{\frac {2(a^{2}x+b^{2}y)}{a^{2}x^{2}+b^{2}y^{2}}}-{\frac {4(a^{4}x^{3}-b^{4}y^{3})}{a^{4}x^{4}-b^{4}y^{4}}}}$

f)  ${\displaystyle {\frac {x}{x^{3}+x^{2}y+xy^{2}+y^{3}}}+{\frac {y}{x^{3}-x^{2}y+xy^{2}-y^{3}}}+{\frac {1}{x^{2}-y^{2}}}-{\frac {1}{x^{2}+y^{2}}}-{\frac {x^{2}+3y^{2}}{x^{4}-y^{4}}}}$

Exercice 2-2

Calculer les sommes algébriques des fractions suivantes :

a)  ${\displaystyle {\frac {8}{(x^{2}+3)(x^{2}-1)}}+{\frac {2}{x^{2}+3}}+{\frac {1}{x+1}}}$

b)  ${\displaystyle {\frac {x^{3}}{x-1}}-{\frac {x^{2}}{x+1}}-{\frac {1}{x-1}}+{\frac {1}{x+1}}}$

c)  ${\displaystyle {\frac {(x+1)^{3}}{x}}-{\frac {(x+1)^{2}}{x+2}}-{\frac {1}{x}}+{\frac {1}{x+2}}}$

d)  ${\displaystyle {\frac {xy}{ab}}+{\frac {(x-a)(y-a)}{a(a-b)}}+{\frac {(x-b)(y-b)}{b(b-a)}}}$

e)  ${\displaystyle {\frac {x^{4}-(x-1)^{2}}{(x+1)^{2}-x^{2}}}+{\frac {x^{2}-(x^{2}-1)^{2}}{x^{2}(x+1)^{2}-1}}+{\frac {x^{2}(x-1)^{2}-1}{x^{4}-(x+1)^{2}}}}$

Exercice 2-3

Calculer et simplifier les sommes algébriques suivantes :

a)  ${\displaystyle {\frac {b+c}{bc}}\left(b^{2}+c^{2}-a^{2}\right)+{\frac {c+a}{ca}}\left(c^{2}+a^{2}-b^{2}\right)+{\frac {a+b}{ab}}\left(a^{2}+b^{2}-c^{2}\right)}$

b)  ${\displaystyle \left({\frac {b}{c}}+{\frac {c}{b}}\right)^{2}+\left({\frac {c}{a}}+{\frac {a}{c}}\right)^{2}+\left({\frac {a}{b}}+{\frac {b}{a}}\right)^{2}-\left({\frac {b}{c}}+{\frac {c}{b}}\right)\left({\frac {c}{a}}+{\frac {a}{c}}\right)\left({\frac {a}{b}}+{\frac {b}{a}}\right)}$

c)  ${\displaystyle {\frac {b-c}{b+c}}+{\frac {c-a}{c+a}}+{\frac {a-b}{a+b}}+{\frac {(b-c)(c-a)(a-b)}{(b+c)(c+a)(a+b)}}}$

d)  ${\displaystyle {\frac {2a}{a+b}}+{\frac {2b}{b+c}}+{\frac {2c}{c+a}}+{\frac {(b-c)(c-a)(a-b)}{(b+c)(c+a)(a+b)}}}$

e)  ${\displaystyle {\frac {bc}{(a+b)(a+c)}}+{\frac {ca}{(b+c)(b+a)}}+{\frac {ab}{(c+a)(c+b)}}+{\frac {2abc}{(b+c)(c+a)(a+b)}}}$

Exercice 2-4

Calculer les sommes algébriques des fractions suivantes :

a)  ${\displaystyle {\frac {a^{3}(b+c)}{(a-b)(a-c}}+{\frac {b^{3}(c+a)}{(b-c)(b-a)}}+{\frac {c^{3}(a+b)}{(c-a)(c-b)}}}$

b)  ${\displaystyle {\frac {(a-x)(a-y)(a-z)}{(a-b)(a-c)}}+{\frac {(b-x)(b-y)(b-z)}{(b-c)(b-a)}}+{\frac {(c-x)(c-y)(c-z)}{(c-a)(c-b)}}}$

c)  ${\displaystyle {\frac {a(b-c)^{2}}{(a-b)(a-c)}}+{\frac {b(c-a)^{2}}{(b-c)(b-a)}}+{\frac {c(a-b)^{2}}{(c-a)(c-b)}}}$

d)  ${\displaystyle {\frac {(x+b)(x+c)}{(a-b)(a-c)}}+{\frac {(x+c)(x+a)}{(b-c)(b-a)}}+{\frac {(x+a)(x+b)}{(c-a)(c-b)}}}$

Exercice 2-5

Calculer les sommes algébriques des fractions suivantes :

a)  ${\displaystyle {\frac {a^{2}(x-b)(x-c)}{(a-b)(a-c)}}+{\frac {b^{2}(x-c)(x-a)}{(b-c)(b-a)}}+{\frac {c^{2}(x-a)(x-b)}{(c-a)(c-b)}}}$

b)  ${\displaystyle {\frac {(x-a)^{2}}{a(a-b)(a-c)}}+{\frac {(x-b)^{2}}{b(b-c)(b-a)}}+{\frac {(x-c)^{2}}{c(c-a)(c-b)}}}$

c)  ${\displaystyle {\frac {x+a}{x(x-y)(x-z)}}+{\frac {y+a}{y(y-z)(y-x)}}+{\frac {z+a}{z(z-x)(z-y)}}}$

d)  ${\displaystyle {\frac {1}{(a-b)(a-c)(x+a)}}+{\frac {1}{(b-c)(b-a)(x+b)}}+{\frac {1}{(c-a)(c-b)(x+c)}}}$