Gra edukacyjna

Polecenie 1

Zagraj w grę, a następnie wykonaj poniższe polecenia.

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Dla $x \in ℝ ∖ ｛ - 1, - 2, 2 ｝$ , $\frac{x - 1}{x^{2} + 3 x + 2} \cdot \frac{x + 2}{x^{2} - x - 2} =$ =1. $\frac{x - 1}{(x + 1) (x - 2)}$ , 2. $\frac{1}{(x - 2) (x + 5)}$ , 3. $\frac{x - 1}{(x + 1) (x + 1) (x - 2) (x - 2)}$ , 4. $\frac{1}{(x + 2) (x - 1)}$ , 5. $\frac{1}{(x + 1) (x + 1)}$ , 6. $\frac{x - 1}{(x + 1) (x + 1) (x - 2)}$ , 7. $\frac{x^{2} + x + 1}{(x - 1) (x + 2) (x - 1)}$ , 8. $\frac{x - 1}{(x + 1) (x + 1)}$ , 9. $\frac{1}{x - 2}$ .

Dla $x \in ℝ ∖ ｛ 2, - 3, - 5, - 2 ｝$ , $\frac{x + 2}{x^{2} + x - 6} \cdot \frac{x + 3}{x^{2} + 7 x + 10} =$ 1. $\frac{x - 1}{(x + 1) (x - 2)}$ , 2. $\frac{1}{(x - 2) (x + 5)}$ , 3. $\frac{x - 1}{(x + 1) (x + 1) (x - 2) (x - 2)}$ , 4. $\frac{1}{(x + 2) (x - 1)}$ , 5. $\frac{1}{(x + 1) (x + 1)}$ , 6. $\frac{x - 1}{(x + 1) (x + 1) (x - 2)}$ , 7. $\frac{x^{2} + x + 1}{(x - 1) (x + 2) (x - 1)}$ , 8. $\frac{x - 1}{(x + 1) (x + 1)}$ , 9. $\frac{1}{x - 2}$ .

Dla $x \in ℝ ∖ ｛ - 2, 3, 1 ｝$ , $\frac{x^{2} + x + 1}{x^{2} - x - 6} \cdot \frac{x - 3}{x^{3} - 1} =$ 1. $\frac{x - 1}{(x + 1) (x - 2)}$ , 2. $\frac{1}{(x - 2) (x + 5)}$ , 3. $\frac{x - 1}{(x + 1) (x + 1) (x - 2) (x - 2)}$ , 4. $\frac{1}{(x + 2) (x - 1)}$ , 5. $\frac{1}{(x + 1) (x + 1)}$ , 6. $\frac{x - 1}{(x + 1) (x + 1) (x - 2)}$ , 7. $\frac{x^{2} + x + 1}{(x - 1) (x + 2) (x - 1)}$ , 8. $\frac{x - 1}{(x + 1) (x + 1)}$ , 9. $\frac{1}{x - 2}$ .

Dla $x \in ℝ ∖ ｛ 1, 2, 3 ｝$ , $\frac{x^{2} + x + 1}{x^{2} - 4 x + 3} \cdot \frac{x - 3}{x^{2} - 3 x + 2} =$ 1. $\frac{x - 1}{(x + 1) (x - 2)}$ , 2. $\frac{1}{(x - 2) (x + 5)}$ , 3. $\frac{x - 1}{(x + 1) (x + 1) (x - 2) (x - 2)}$ , 4. $\frac{1}{(x + 2) (x - 1)}$ , 5. $\frac{1}{(x + 1) (x + 1)}$ , 6. $\frac{x - 1}{(x + 1) (x + 1) (x - 2)}$ , 7. $\frac{x^{2} + x + 1}{(x - 1) (x + 2) (x - 1)}$ , 8. $\frac{x - 1}{(x + 1) (x + 1)}$ , 9. $\frac{1}{x - 2}$ .

Dla $x \in ℝ ∖ ｛ - 1, - 2, 2 ｝$ , $\frac{x - 1}{x^{3} + x^{2} - 4 x - 4} \cdot \frac{x + 2}{x^{2} - x - 2} =$ 1. $\frac{x - 1}{(x + 1) (x - 2)}$ , 2. $\frac{1}{(x - 2) (x + 5)}$ , 3. $\frac{x - 1}{(x + 1) (x + 1) (x - 2) (x - 2)}$ , 4. $\frac{1}{(x + 2) (x - 1)}$ , 5. $\frac{1}{(x + 1) (x + 1)}$ , 6. $\frac{x - 1}{(x + 1) (x + 1) (x - 2)}$ , 7. $\frac{x^{2} + x + 1}{(x - 1) (x + 2) (x - 1)}$ , 8. $\frac{x - 1}{(x + 1) (x + 1)}$ , 9. $\frac{1}{x - 2}$ .

Dla $x \in ℝ ∖ ｛ - 1, - 2, 2, 5 ｝$ , $\frac{x - 5}{x^{3} - 2 x^{2} - 13 x - 10} \cdot \frac{x + 2}{x - 2} =$ 1. $\frac{x - 1}{(x + 1) (x - 2)}$ , 2. $\frac{1}{(x - 2) (x + 5)}$ , 3. $\frac{x - 1}{(x + 1) (x + 1) (x - 2) (x - 2)}$ , 4. $\frac{1}{(x + 2) (x - 1)}$ , 5. $\frac{1}{(x + 1) (x + 1)}$ , 6. $\frac{x - 1}{(x + 1) (x + 1) (x - 2)}$ , 7. $\frac{x^{2} + x + 1}{(x - 1) (x + 2) (x - 1)}$ , 8. $\frac{x - 1}{(x + 1) (x + 1)}$ , 9. $\frac{1}{x - 2}$ .

Dla $x \in ℝ ∖ ｛ - 1, - 2, 2 ｝$ , $\frac{x^{2} - 3 x + 2}{x^{2} + 3 x + 2} \cdot \frac{x + 2}{x^{2} - x - 2} =$ 1. $\frac{x - 1}{(x + 1) (x - 2)}$ , 2. $\frac{1}{(x - 2) (x + 5)}$ , 3. $\frac{x - 1}{(x + 1) (x + 1) (x - 2) (x - 2)}$ , 4. $\frac{1}{(x + 2) (x - 1)}$ , 5. $\frac{1}{(x + 1) (x + 1)}$ , 6. $\frac{x - 1}{(x + 1) (x + 1) (x - 2)}$ , 7. $\frac{x^{2} + x + 1}{(x - 1) (x + 2) (x - 1)}$ , 8. $\frac{x - 1}{(x + 1) (x + 1)}$ , 9. $\frac{1}{x - 2}$ .

Dla $x \in ℝ ∖ ｛ 1, - 2, 2, 3 ｝$ , $\frac{x + 3}{x^{2} + x - 2} \cdot \frac{x^{2} + x - 2}{x^{2} + x - 6} =$ 1. $\frac{x - 1}{(x + 1) (x - 2)}$ , 2. $\frac{1}{(x - 2) (x + 5)}$ , 3. $\frac{x - 1}{(x + 1) (x + 1) (x - 2) (x - 2)}$ , 4. $\frac{1}{(x + 2) (x - 1)}$ , 5. $\frac{1}{(x + 1) (x + 1)}$ , 6. $\frac{x - 1}{(x + 1) (x + 1) (x - 2)}$ , 7. $\frac{x^{2} + x + 1}{(x - 1) (x + 2) (x - 1)}$ , 8. $\frac{x - 1}{(x + 1) (x + 1)}$ , 9. $\frac{1}{x - 2}$ .

Dla $x \in ℝ ∖ ｛ - 1, - 2, 5, 2 ｝$ , $\frac{x - 5}{x^{2} + 3 x + 2} \cdot \frac{x^{2} + x - 2}{x^{2} - 7 x + 10} =$ 1. $\frac{x - 1}{(x + 1) (x - 2)}$ , 2. $\frac{1}{(x - 2) (x + 5)}$ , 3. $\frac{x - 1}{(x + 1) (x + 1) (x - 2) (x - 2)}$ , 4. $\frac{1}{(x + 2) (x - 1)}$ , 5. $\frac{1}{(x + 1) (x + 1)}$ , 6. $\frac{x - 1}{(x + 1) (x + 1) (x - 2)}$ , 7. $\frac{x^{2} + x + 1}{(x - 1) (x + 2) (x - 1)}$ , 8. $\frac{x - 1}{(x + 1) (x + 1)}$ , 9. $\frac{1}{x - 2}$ .

Polecenie 2

Oblicz $\frac{4 x - 12}{x^{2} - 6 x + 9} \cdot \frac{x^{2} + 2 x - 15}{6 x^{2} - 150}$ .

Pamiętaj o skracaniu.
Określ dziedzinę.

$\frac{4 x - 12}{x^{2} - 6 x + 9} \cdot \frac{x^{2} + 2 x - 15}{6 x^{2} - 150} =$
$= \frac{4 (x - 3)}{{(x - 3)}^{2}} \cdot \frac{(x - 3) (x + 5)}{6 (x - 5) (x + 5)}$

$\frac{4 x - 12}{x^{2} - 6 x + 9} \cdot \frac{x^{2} + 2 x - 15}{6 x^{2} - 150} =$
$= \frac{4 (x - 3)}{(x - 3)^{2}} \cdot \frac{(x - 3) (x + 5)}{6 (x - 5) (x + 5)} =$
$= \frac{4}{6 (x - 5)} = \frac{2}{3 (x - 5)}$

Polecenie 3

Oblicz $\frac{x^{2} - 5 x - 6}{4 x - 24} \cdot \frac{6 x - 6}{x^{2} - 1}$ .

Pamiętaj o skracaniu.
Określ dziedzinę.

$\frac{x^{2} - 5 x - 6}{4 x - 24} \cdot \frac{6 x - 6}{x^{2} - 1} =$
$= \frac{(x - 6) (x + 1)}{4 (x - 6)} \cdot \frac{6 (x - 1)}{(x + 1) (x - 1)}$

$\frac{x^{2} - 5 x - 6}{4 x - 24} \cdot \frac{6 x - 6}{x^{2} - 1} =$
$= \frac{(x - 6) (x + 1)}{4 (x - 6)} \cdot \frac{6 (x - 1)}{(x + 1) (x - 1)} =$
$= \frac{6}{4} = \frac{3}{2}$

Polecenie 4

Oblicz $\frac{15 x^{2} - 4 x - 4}{9 x^{2} - 4} \cdot \frac{x^{2} - x - 12}{25 x^{2} - 4}$ .

Pamiętaj o skracaniu.
Określ dziedzinę.

$\frac{15 x^{2} - 4 x - 4}{9 x^{2} - 4} \cdot \frac{x^{2} - x - 12}{25 x^{2} - 4} =$
$= \frac{(5 x + 2) (3 x - 2)}{(3 x - 2) (3 x + 2)} \cdot \frac{(x + 3) (x - 4)}{(5 x + 2) (5 x - 2)}$

$\frac{15 x^{2} - 4 x - 4}{9 x^{2} - 4} \cdot \frac{x^{2} - x - 12}{25 x^{2} - 4} =$
$= \frac{(5 x + 2) (3 x - 2)}{(3 x - 2) (3 x + 2)} \cdot \frac{(x + 3) (x - 4)}{(5 x + 2) (5 x - 2)} =$
$= \frac{(x + 3) (x - 4)}{(3 x + 2) (5 x - 2)}$

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