Vidéo de théorie et exercices-Multiplication et division de fractions algébriques
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a) \(\frac {x^2 - 9}{x^2 + 10x + 25} \times \frac {x + 5}{x + 3}\) b) \(\frac {3x^2 + 11x - 4}{12x^2 - 4x} \times \frac {4x^2 - 28x + 40}{x^2 - x -20}\) c) \(\frac {x^2 - 25}{x^2 + 7x + 10} \div \frac {xy - 5y}{xy + 2y}\) d) \(\frac {x^2 - 25}{2x^2 + 7x + 3} \div \frac {x^2 + 10x + 25}{x^2 - 9}\) e) \(\frac {2y^2 - 5y + 2}{2y^2 + 5y - 3} \times \frac {2y^2 + 7y + 3}{4 - y^2} \div \frac {2y^2 + 5y + 2}{3y^2 - 7y + 2}\) f) \(\frac {5x + 30}{2x + 5} \times \frac {x^2 - 16}{x^2 + 8x + 16} \div \frac {x - 4}{4x^2 + 10x}\) g) \(\frac {3x - 2}{x - 2} \div \frac {9x^2 - 12x + 4}{x^2 - 4}\) h) \(\frac {(x^2 - y^2)^2}{(x - y)^2} \div \frac {(x^4 - y^4)}{(x + y)^2} \times \frac {(x^2 + y^2)}{(x + y)^3}\) |
a)
\(\frac {x^2 - 9}{x^2 + 10x + 25} \times \frac {x + 5}{x + 3} = \)
\(\frac {(x + 3)(x - 3)}{(x + 5)(x + 5)} \times \frac {x + 5}{x + 3} = \)
\(\frac {x - 3}{x + 5}\)
restrictions : \(x \neq -5\) et \(x \neq -3\)
b)
\(\frac {3x^2 + 11x - 4}{12x^2 - 4x} \times \frac {4x^2 - 28x + 40}{x^2 - x -20} = \)
\(\frac {(x + 4)(3x - 1)}{4x(3x - 1)} \times \frac {(x - 5)(4x - 8)}{(x - 5)(x + 4)} = \)
\(\frac {(x + 4)}{4x} \times \frac {(4x - 8)}{(x + 4)} = \)
\(\frac {4(x - 2)}{4x} = \)
\(\frac {(x - 2)}{x}\)
restrictions : \(x \neq 0\), \(x \neq \frac {1}{3}\), \(x \neq -4\) et \(x \neq 5\)
c)
\(\frac {x^2 - 25}{x^2 + 7x + 10} \div \frac {xy - 5y}{xy + 2y} = \)
\(\frac {(x + 5)(x - 5)}{(x + 2)(x + 5)} \div \frac {y(x - 5)}{y(x + 2)} = \)
\(\frac {(x - 5)}{(x + 2)} \div \frac {(x - 5)}{(x + 2)} = \)
\(\frac {(x - 5)}{(x + 2)} \times \frac {(x + 2)}{(x - 5)}\)
\(1\)
restrictions : \(x \neq \pm5\), \(x \neq -2\) et \(y \neq 0\)
d)
\(\frac {x^2 - 25}{2x^2 + 7x + 3} \div \frac {x^2 + 10x + 25}{x^2 - 9} = \)
\(\frac {(x - 5)(x + 5)}{(x + 3)(2x + 1)} \div \frac {(x + 5)(x + 5)}{(x - 3)(x + 3)} = \)
\(\frac {(x - 5)(x + 5)}{(x + 3)(2x + 1)} \times \frac {(x - 3)(x + 3)}{(x + 5)(x + 5)} = \)
\(\frac {(x - 5)}{(2x + 1)} \times \frac {(x - 3)}{(x + 5)} = \)
\(\frac {(x - 5)(x - 3)}{(2x + 1)(x + 5)}\)
restrictions : \(x \neq -5\), \(x \neq \pm 3\) et \(x \neq \frac {-1}{2}\)
e)
\(\frac {2y^2 -5y + 2}{2y^2 + 5y - 3} \times \frac {2y^2 + 7y + 3}{4 - y^2} \div \frac {2y^2 + 5y + 2}{3y^2 - 7y + 2} = \)
\(\frac {(y - 2)(2y - 1)}{(y + 3)(2y - 1)} \times \frac {(y + 3)(2y + 1)}{(2 - y)(2 + y)} \div \frac {(y + 2)(2y + 1)}{(y - 2)(3y - 1)} = \)
\(\frac {(y - 2)}{1} \times \frac {(2y + 1)}{(2 - y)(2 + y)} \div \frac {(y + 2)(2y + 1)}{(y - 2)(3y - 1)} = \)
\(\frac {(y - 2)(2y + 1)}{(2 - y)(2 + y)} \div \frac {(y + 2)(2y + 1)}{(y - 2)(3y - 1)} = \)
\(\frac {(y - 2)(2y + 1)}{-1(y-2)( y+2)} \times \frac {(y - 2)(3y - 1)}{(y + 2)(2y + 1)} = \)
\(\frac {-(3y - 1)(y-2)}{( y+2)(y + 2)}\)
restrictions : \(y \neq -3\), \(y \neq \pm 2\), \(y \neq \frac {1}{3}\) et \(y \neq \frac {1}{2}\), \(y \neq \frac {-1}{2}\)
f)
\(\frac {5x + 30}{2x + 5} \times \frac {x^2 - 16}{x^2 + 8x + 16} \div \frac {x - 4}{4x^2 + 10x} = \)
\(\frac {5(x + 6)}{(2x + 5)} \times \frac {(x + 4)(x - 4)}{(x + 4)(x + 4)} \div \frac {(x - 4)}{2x(2x + 5)} = \)
\(\frac {5(x + 6)}{2x + 5} \times \frac {(x - 4)}{(x + 4)} \div \frac {(x - 4)}{2x(2x + 5)} = \)
\(\frac {5(x + 6)}{2x + 5} \times \frac {(x - 4)}{(x + 4)} \times \frac {2x(2x + 5)}{(x - 4)} = \)
\(\frac {5(x + 6)}{1} \times \frac {1}{(x + 4)} \times \frac {2x}{1} = \)
\(\frac {5(x + 6) \times 2x}{(x + 4)} = \)
\(\frac {10x(x + 6)}{(x + 4)}\)
restrictions : \(x \neq 0\), \(x \neq\pm4\) et \(x \neq \frac {-5}{2}\)
g)
\(\frac {3x - 2}{x - 2} \div \frac {9x^2 - 12x + 4}{x^2 - 4} = \)
\(\frac {(3x - 2)}{(x - 2)} \div \frac {(3x - 2)(3x - 2)}{(x + 2)(x - 2)} = \)
\(\frac {(3x - 2)}{(x - 2)} \times \frac {(x + 2)(x - 2)}{(3x - 2)(3x - 2)} = \)
\(\frac {1}{1} \times \frac {(x + 2)}{(3x - 2)} = \)
\(\frac {(x + 2)}{(3x - 2)}\)
restrictions : \(x \neq \pm 2\) et \(x \neq \frac {2}{3}\)
h)
\(\frac {(x^2 - y^2)^2}{(x - y)^2} \div \frac {(x^4 - y^4)}{(x + y)^2} \times \frac {(x^2 + y^2)}{(x + y)^3} = \)
\(\frac {(x + y)(x - y)(x + y)(x - y)}{(x - y)(x - y)} \div \frac {(x^2 + y^2)(x - y)(x + y)}{(x + y)(x + y)} \times \frac {(x^2 + y^2)}{(x + y)(x + y)(x + y)} = \)
\(\frac {(x + y)(x + y)}{1} \div \frac {(x^2 + y^2)(x - y)}{(x + y)} \times \frac {(x^2 + y^2)}{(x + y)(x + y)(x + y)} = \)
\(\frac {(x + y)(x + y)}{1} \times \frac {(x + y)}{(x^2 + y^2)(x - y)} \times \frac {(x^2 + y^2)}{(x + y)(x + y)(x + y)} = \)
\(\frac {1}{1} \times \frac {1}{(x - y)} \times \frac {1}{1} = \)
\(\frac {1}{x - y}\), restriction : \( x \neq \pm y\)
Multiplication de fractions algébriques
Prendre note d'une petite erreur de calcul à 1:16. La réponse est 15/14 et non 15/10.
Division de fractions algébriques.