Vidéo de théorie et exercices-Fractions algébriques
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Attention! Petite coquille dans la vidéo, à l'exemple E. On devrait lire a²+5a+ 6 et non a²+5x+ 6. |
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Simplifie les fractions rationnelles suivantes : a) \(\frac {(ax + ay)}{(bx + by)}\) b) \(\frac {(5x - 15)}{(4x - 12)}\) c) \(\frac {(2x^2 + x)}{(6x + 3)}\) d) \(\frac {(15x + 3)}{(30x^2 + 6x)}\) e) \(\frac {(12x^3 + 8x^2)}{(6x^2 + 4x)}\) |
a)
\(\frac {a(x + y)}{b(x + y)} = \frac {a}{b}\), si \(b \neq 0\) et \(x \neq -y\)
b)
\(\frac {5(x - 3)}{4(x - 3)} = \frac {5}{4}\), si \(x \neq 3\)
c)
\(\frac {x(2x + 1)}{3(2x + 1)} = \frac {x}{3}\), si \(x \neq \frac {-1}{2}\)
d)
\(\frac {3(5x + 1)}{6x(5x + 1)} = \frac {3}{6x} = \frac {1}{2x}\), si \(x \neq 0\) et \(x \neq \frac {-1}{5}\)
e)
\(\frac {4x^2(3x + 2)}{2x(3x + 2)} = 2x\), si \(x \neq 0\) et \(x \neq \frac {-2}{3}\)
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Simplifie les fractions algébriques suivantes : a) \(\frac {ax + 3x + ay + 3y}{x + y}\) b) \(\frac {4a^2 - 3a - 4ab + 3b}{12a - 9}\) c) \(\frac {5ax - 2bx + 5a - 2b}{15ax - 6bx - 10a + 4b}\) d) \(\frac {x(x + 4) - 3(x + 4)}{4x - 12}\) e) \(\frac {2ax^2 + 3ax + 4x^2 + 6x}{2a^2x + 3a^2 + 4ax + 6a}\) |
a)
\(\frac {(x + y)(a + 3)}{(x + y)} = (a + 3)\) si \(x \neq -y\)
b)
\(\frac {(a - b)(4a - 3)}{3(4a - 3)} = \frac {(a - b)}{3}\), si \(a \neq \frac {3}{4}\)
c)
\(\frac {(x + 1)(5a - 2b)}{(5a - 2b)(3x - 2)} = \frac {(x + 1)}{(3x - 2)}\), si \(5a \neq 2b\) et \(x \neq \frac {2}{3}\)
d)
\(\frac {(x + 4)(x - 3)}{4(x - 3)} = \frac {(x + 4)}{4}\), si \(x \neq 3\)
e)
\(\frac {(2x + 3)(ax + 2x)}{(2x + 3)(a^2 + 2a)} = \frac {(ax + 2x)}{(a^2 + 2a)} = \frac {x(a + 2)}{a(a + 2)} = \frac {x}{a}\), si \(a \neq 0\), \(a \neq -2\) et \(x \neq \frac {-3}{2}\)
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Simplifie les fractions algébriques suivantes : a) \(\frac {a^2 -9}{2a + 6}\) b) \(\frac {9a^2 + 6a}{9a^2 - 4}\) c) \(\frac {x^2 - 4x}{x^2 - 16}\) d) \(\frac {x^2 - 1}{(2x^2 - 2x)(x + 1)}\) e) \(\frac {x^2 - 1}{(x + 1)^2}\) |
a) \(\frac {(a + 3)(a - 3)}{2(a + 3)} = \frac {(a - 3)}{2}\), si \(a \neq -3\)
b) \(\frac {3a(3a + 2)}{(3a + 2)(3a - 2)} = \frac {3a}{(3a - 2)}\), si \(a \neq \pm \frac {2}{3}\)
c) \(\frac {x(x - 4)}{(x + 4)(x - 4)} = \frac {x}{(x + 4)}\), si \(x \neq \pm 4\)
d) \(\frac {(x + 1)(x - 1)}{2x(x - 1)(x + 1)} = \frac {1}{2x}\), si \(x \neq 0\) et \(x \neq \pm 1\)
e) \(\frac {(x + 1)(x - 1)}{(x + 1)^2} = \frac {(x - 1)}{(x + 1)}\), si \(x \neq -1\)
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Simplifie les fractions algébriques suivantes : a) \(\frac {x^2 - x - 30}{x^2 + 9x + 20}\) b) \(\frac {a^2 + 8ab - 9b^2}{a^2 - 2ab + b^2}\) c) \(\frac {a^2 - 25}{a^2 + 10a + 25}\) d) \(\frac {x^2 - 5x + 4}{x^4 - 17x^2 + 16}\) e) \(\frac {(x^2 - 6x + 5)(x^2 + 6x + 5)}{2x^2 - 50}\) |
a)
\(\frac {(x - 6)(x + 5)}{(x + 4)(x + 5)} = \frac {(x - 6)}{(x + 4)}\), si \(x \neq -4\) et \(x \neq -5\)
b)
\(\frac {(a + 9b)(a - b)}{(a - b)(a - b)} = \frac {(a + 9b)}{(a - b)}\), si \(a \neq b\)
c)
\(\frac {(a + 5)(a - 5)}{(a + 5)(a + 5)} = \frac {(a - 5)}{(a + 5)}\), si \(a \neq -5\)
d)
\(\frac {(x - 4)(x - 1)}{(x + 4)(x - 4)(x + 1)(x - 1)} = \frac {1}{(x + 4)(x + 1)}\), si \(x \neq \pm 1\) et \(x \neq \pm 4\)
e)
\(\frac {(x - 5)(x - 1)(x + 5)(x + 1)}{2(x + 5)(x - 5)} = \frac {(x - 1)(x + 1)}{2}\), si \(x \neq \pm 5\)
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Simplifie les fractions algébriques suivantes : a) \(\frac {x^2 - 1}{x^2 - 2x + 1}\) b) \(\frac {a^2 + 5a + 6}{a^2 + 6a + 8}\) c) \(\frac {2a^2 + 2a - 24}{10a^2 - 160}\) d) \(\frac {ab + bc + a + c}{ab - bc + a - c}\) e) \(\frac {2x^2 - 32x + 128}{4x^2 - 256}\) f) \(\frac {2x^2 + 17x + 21}{3x^2 + 26x + 35}\) g) \(\frac {(x^2 - 16)(x^2 - 9)}{x^2 + 7x + 12}\) h) \(\frac {(x^2 - 1)(x^2 + 1)}{2x^4 - 4x^2 + 2}\) i) \(\frac {(x^2 - 7x + 10)(x^2 - 25)}{(x^2 - 2x - 15)(x^2 - 4)}\) j) \(\frac {(x^2 + 2xy + y^2)(x^2 - 2xy + y^2)}{x^2 - y^2}\) |
a)
\(\frac {(x + 1)(x - 1)}{(x - 1)(x - 1)} = \frac {(x + 1)}{(x - 1)}\), si \(x \neq 1\)
b)
\(\frac {(a + 3)(a + 2)}{(a + 2)(a + 4)} = \frac {(a + 3)}{(a + 4)}\), si \(a \neq -4\) et \(a \neq -2\)
c)
\(\frac {2(a + 4)(a - 3)}{10(a + 4)(a - 4)} = \frac {2(a - 3)}{10(a - 4)} = \frac {(a - 3)}{5(a - 4)}\), si \(a \neq \pm 4\)
d)
\(\frac {(a + c)(b + 1)}{(b + 1)(a - c)} = \frac {(a + c)}{(a - c)}\), si \(b \neq -1\) et \(a \neq c\)
e)
\(\frac {2(x - 8)(x - 8)}{4(x - 8)(x + 8)} = \frac {2(x - 8)}{4(x + 8)} = \frac {(x - 8)}{2(x + 8)}\), si \(x \neq \pm 8\)
f)
\(\frac {(x + 7)(2x + 3)}{(x + 7)(3x + 5)} = \frac {(2x + 3)}{(3x + 5)}\), si \(x \neq -7\) et \(x \neq \frac {-5}{3}\)
g)
\(\frac {(x + 4)(x - 4)(x + 3)(x - 3)}{(x + 4)(x + 3)} = (x - 4)(x - 3)\), si \(x \neq -4\) et \(x \neq -3\)
h)
\(\frac {(x + 1)(x - 1)(x^2 + 1)}{2(x^2 - 1)^2} = \frac {(x^2 + 1)}{2(x + 1)(x-1)}\) et \(x \neq \pm1\)
i)
\(\frac {(x - 5)(x - 2)(x - 5)(x + 5)}{(x - 5)(x + 3)(x + 2)(x - 2)} = \frac {(x - 5)(x + 5)}{(x + 3)(x + 2)}\), si \(x \neq \pm 2\), \(x \neq 5\) et \(x \neq -3\)
j)
\(\frac {(x + y)(x + y)(x - y)(x - y)}{(x + y)(x - y)} = (x + y)(x - y)\), si \(x \neq \pm y\)