Question

25 puncte
Mentionati conditiile in care fractia din poza este definita si simplificati-o.... ​Pas cu pas...

Answer 1

 

$\displaystyle\bf\\\text{Fractia are sens doar daca la numitor este: }~5x^3-10x^2-15x\\\\\frac{5x^3-45x}{5x^3-10x^2-15x}\\\\\textbf{Numitorul trebuie sa fie diferit de zero}\\5x^3-10x^2-15x=\\=5x[x^2+x-3x-3]=\\=5x[x(x+1)-3(x+1)]=\\=5x(x+1)(x-3)\\~~~~~5x=0\implies~x_1=0\\x+1=0\implies~x_2=-1\\x-3=0\implies~x_3=3\\\\Contitiile~sunt:~~x\in R-\{0;~-1;~3\}$

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$\displaystyle\bf\\\textbf{Simplificam fractia:}\\\\\frac{5x^3-45x}{5x^3-10x^2-15x}=\\\\\\=\frac{5x(x^2-9)}{5x(x+1)(x-3)}=\\\\\\=\frac{5x(x+3)(x-3)}{5x(x+1)(x-3)}=~~~~~Simplificam~cu~~5x~~si~cu~~(x-3)\\\\\\=\boxed{\bf\frac{(x+3)}{(x+1)}}$