diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index a966f82..89b4cb5 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -8,17 +8,17 @@ "type": "tabs", "children": [ { - "id": "c4502a3f6b7de95c", + "id": "f9decbc6fa574701", "type": "leaf", "state": { "type": "markdown", "state": { - "file": "notes/NIEMIECKI 19.05.2025.md", + "file": "mathematika/wielomiany/Wielomiany notatki.md", "mode": "preview", "source": false }, "icon": "lucide-file", - "title": "NIEMIECKI 19.05.2025" + "title": "Wielomiany notatki" } } ] @@ -78,8 +78,7 @@ } ], "direction": "horizontal", - "width": 300, - "collapsed": true + "width": 300 }, "right": { "id": "8a799a669a1bb56d", @@ -249,9 +248,14 @@ }, "active": "801cb97c4f56dcb9", "lastOpenFiles": [ + "mathematika/wielomiany/Drawing 2025-05-19 17.00.15.excalidraw.md", + "mathematika/wielomiany/Wielomiany notatki.md", + "notes/NIEMIECKI 19.05.2025.md", + "mathematika/wielomiany/Untitled.canvas", + "mathematika/wielomiany", + "mathematika", "notes/POLSKI 16.05.2025.md", "notes/HISTORIA 16.05.2025.md", - "notes/NIEMIECKI 19.05.2025.md", "conflict-files-obsidian-git.md", "Excalidraw/Drawing 2025-05-18 18.45.08.excalidraw.md", "Excalidraw/Drawing 2025-05-18 18.39.11.excalidraw.md", diff --git a/mathematika/wielomiany/Drawing 2025-05-19 17.00.15.excalidraw.md b/mathematika/wielomiany/Drawing 2025-05-19 17.00.15.excalidraw.md new file mode 100644 index 0000000..deb6ac3 --- /dev/null +++ b/mathematika/wielomiany/Drawing 2025-05-19 17.00.15.excalidraw.md @@ -0,0 +1,3436 @@ +--- + +excalidraw-plugin: parsed +tags: [excalidraw] + +--- +==⚠ Switch to EXCALIDRAW VIEW in the MORE OPTIONS menu of this document. ⚠== You can decompress Drawing data with the command palette: 'Decompress current Excalidraw file'. For more info check in plugin settings under 'Saving' + + +# Excalidraw Data + +## Text Elements +%% +## Drawing +```compressed-json +N4KAkARALgngDgUwgLgAQQQDwMYEMA2AlgCYBOuA7hADTgQBuCpAzoQPYB2KqATLZMzYBXUtiRoIACyhQ4zZAHoFAc0JRJQgEYA6bGwC2CgF7N6hbEcK4OCtptbErHALRY8RMpWdx8Q1TdIEfARcZgRmBShcZQUebQBGeISaOiCEfQQOKGZuAG1wMFAwYogSbggAEQAJAAUAKwBHACEAcQA5QgAWAGVugE02qvwAUUJ4gEkmlOLIWERygDNAhE8q + +fhLMbmcAVgBOAHZtfe31yBgtgGZ93e0ebf2ADni704gKEnVuHgAGV8kEQjKaTcC4nAqQazKYLcX7giDMKCkNgAawQAGE2Pg2KRygBieIIAkE6YlTS4bDI5RIoQcYgYrE4iSI6zMOC4QJZEmQBaEfD4bqwaESQQeLnwxEohAAdQ+km48VeCKRqIFMCF6BFZVe1KBHHCOTQCrhbDZ2DU50N31hMwgVOEcHGxANqFyAF1XktcBlHdwOEI+a9CLSsOVc + +N8xdTaXrmM6/QG4WEVvKHg8eAA2XbfTr7PhwxgsdhcNDZ1751icNqcMTcXZPeJ7C4PTqB5gVNJQJNoBYEMKvTTCWnDYIZLLOt2vIRwYi4DvEeX7bNptMPXZp+Kgo02ogcZG+/34V5Yimd1Dd/C9uFwNhB7J5cFgfIzYrWp/fe/u++Pp/OTev9+nB97zAH8ALAeJ/0/ICQKAngIKfL8ZmcXMnzAWCnw/eCoN/GY0JmDCZgQ59QNw4p8OKQjUOIuCC + 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b/mathematika/wielomiany/Wielomiany notatki.md @@ -0,0 +1,329 @@ +## **2.1. Stopień i współczynniki wielomianu** + +### **Co to jest wielomian?** + +Wyobraź sobie, że wielomian to **torba z zakupami**. W środku masz różne produkty: jabłka (`x`), banany (`x²`), marchewki (`x³`), itd. Każdy produkt ma **swoją cenę** (współczynnik) i **ilość** (potęga `x`). + +**Przykład:** +`5x³ - 2x² + 7x - 1` + +- `5x³` → 5 worków jabłek (`x³`), każdy kosztuje 5 zł. + +- `-2x²` → 2 banany (`x²`), ale na minusie (ktoś je zabrał). + +- `7x` → 7 marchewek (`x`). + +- `-1` → ktoś ukradł 1 zł z portfela (wyraz wolny). + + +### **Stopień wielomianu** + +To **najwyższa potęga** w torbie. W `5x³ - 2x² + 7x - 1` najwyższa to `x³`, więc stopień to **3**. + +### **Współczynniki** + +To liczby przed `x`: + +- przy `x³` → `5` + +- przy `x²` → `-2` + +- przy `x` → `7` + +- wyraz wolny → `-1` + + +**Ćwiczenie:** W `-x⁴ + 3x - 5` stopień to **4**, współczynniki: `-1` (przy `x⁴`), `3` (przy `x`), `-5` (wyraz wolny). + +--- + +## **2.2. Dodawanie i odejmowanie wielomianów** + +### **Dodawanie** + +To jak **łączenie dwóch toreb zakupów**. Możesz dodać tylko **te same produkty** (te same potęgi `x`). + +**Przykład:** +`(3x² + 2x + 1) + (x² - 5x + 4)` + +1. Łączymy `x²`: `3x² + x² = 4x²` + +2. Łączymy `x`: `2x - 5x = -3x` + +3. Łączymy wolne wyrazy: `1 + 4 = 5` + **Wynik:** `4x² - 3x + 5` + + +### **Odejmowanie** + +Najpierw **zmieniamy znaki w drugim nawiasie**, potem dodajemy. + +**Przykład:** +`(3x² + 2x) - (x² - 5x)` + +1. Zmieniamy znaki: `-x² + 5x` + +2. Dodajemy: `3x² - x² + 2x + 5x = 2x² + 7x` + + +--- + +## **2.3. Mnożenie wielomianów** + +To jak **rozdawanie prezentów każdemu po kolei**. Każdy element z pierwszego nawiasu mnożymy przez każdy z drugiego. + +**Przykład:** +`(x + 2)(x + 3)` + +1. `x * x = x²` + +2. `x * 3 = 3x` + +3. `2 * x = 2x` + +4. `2 * 3 = 6` + Teraz dodajemy wszystko: `x² + 3x + 2x + 6 = x² + 5x + 6` + + +**Wizualizacja:** + +(x + 2)(x + 3) = x * (x + 3) + 2 * (x + 3) += x² + 3x + 2x + 6 += x² + 5x + 6 + +--- + +## **2.4. Wzory skróconego mnożenia** + +To **gotowe przepisy** na szybkie mnożenie. + +### **1. Kwadrat sumy** + +`(a + b)² = a² + 2ab + b²` +**Przykład:** +`(x + 1)² = x² + 2*x*1 + 1² = x² + 2x + 1` + +### **2. Kwadrat różnicy** + +`(a - b)² = a² - 2ab + b²` +**Przykład:** +`(x - 2)² = x² - 4x + 4` + +### **3. Różnica kwadratów** + +`(a + b)(a - b) = a² - b²` +**Przykład:** +`(x + 3)(x - 3) = x² - 9` + +--- + +## **Trójkąt Pascala** + +To **piramida liczb**, która pokazuje współczynniki przy `(a + b)ⁿ`. + +Copy + +Download + +1 ← (a + b)⁰ = 1 +1 1 ← (a + b)¹ = a + b +1 2 1 ← (a + b)² = a² + 2ab + b² +1 3 3 1 ← (a + b)³ = a³ + 3a²b + 3ab² + b³ +1 4 6 4 1 ← (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴ + +**Jak czytać?** + +- Dla `(a + b)³` patrzysz na 4. linię: `1 3 3 1` → `a³ + 3a²b + 3ab² + b³`. + + +--- + +## **2.5-2.6. Rozkład wielomianu na czynniki** + +To **odwrotność mnożenia**. Szukamy, z jakich nawiasów mógł powstać wielomian. + +### **Metody rozkładu:** + +1. **Wyciąganie wspólnego czynnika przed nawias** + `6x² + 3x = 3x(2x + 1)` + +2. **Wzory skróconego mnożenia** + `x² - 9 = (x - 3)(x + 3)` + +3. **Grupowanie wyrazów** + `x³ + x² + x + 1 = x²(x + 1) + 1(x + 1) = (x² + 1)(x + 1)` + +4. **Szukanie pierwiastków** (o tym później). + + +--- + +## **2.7. Równania wielomianowe** + +To gdy wielomian **przyrównujemy do zera** i szukamy `x`. + +**Przykład:** +`x² - 5x + 6 = 0` + +1. Rozkładamy na czynniki: `(x - 2)(x - 3) = 0` + +2. Każdy nawias może dać zero: + + - `x - 2 = 0` → `x = 2` + + - `x - 3 = 0` → `x = 3` + **Rozwiązania:** `x = 2` lub `x = 3`. + + +--- + +## **2.8. Dzielenie wielomianów** + +Działa jak **dzielenie pisemne**, ale z `x`. + +**Przykład:** +`(x² + 5x + 6) : (x + 2)` + +1. Dzielimy **pierwszy wyraz** przez pierwszy wyraz dzielnika: + `x² : x = x` → zapisujemy `x` na górze. + +2. Mnożymy `x * (x + 2) = x² + 2x` i odejmujemy od wielomianu: + `(x² + 5x + 6) - (x² + 2x) = 3x + 6` + +3. Powtarzamy: `3x : x = 3` → `3` dopisujemy na górze. + +4. Mnożymy `3 * (x + 2) = 3x + 6` i odejmujemy: + `(3x + 6) - (3x + 6) = 0` → **reszta 0**. + **Wynik:** `x + 3`. + + +--- + +## **2.10. Twierdzenie Bézouta** + +Mówi: **Jeśli `W(a) = 0`, to `(x - a)` jest dzielnikiem wielomianu `W(x)`.** + +**Przykład:** +`W(x) = x² - 4` +Sprawdzamy `W(2) = 2² - 4 = 0` → `(x - 2)` jest dzielnikiem. +Faktycznie: `x² - 4 = (x - 2)(x + 2)`. + +--- + +## **2.11. Pierwiastki całkowite i wymierne** + +### **Pierwiastki całkowite** + +To **całkowite liczby**, które podstawione za `x` dają `0`. +**Jak szukać?** Sprawdzamy dzielniki wyrazu wolnego. + +**Przykład:** +`x³ - 3x² - 4x + 12 = 0` +Dzielniki `12`: `±1, ±2, ±3, ±4, ±6, ±12`. +Sprawdzamy: + +- `W(2) = 8 - 12 - 8 + 12 = 0` → `x = 2` jest pierwiastkiem. + + +### **Pierwiastki wymierne** + +To **ułamki** `p/q`, gdzie `p` to dzielnik wyrazu wolnego, a `q` dzielnik współczynnika przy najwyższej potędze. + +**Przykład:** +`2x² - x - 1 = 0` +Możliwe ułamki: `±1, ±1/2`. +Sprawdzamy `x = 1`: `2 - 1 - 1 = 0` → `x = 1` jest pierwiastkiem. + +--- + +## **2.12. Pierwiastki wielokrotne** + +To gdy **ta sama liczba** jest pierwiastkiem **kilka razy**. + +**Przykład:** +`(x - 1)² = 0` → `x = 1` jest **podwójnym** pierwiastkiem. + +--- + +## **2.13. Wykres wielomianu** + +- **Stopień 1 (np. `2x + 1`)** → prosta linia. + +- **Stopień 2 (np. `x²`)** → parabola (kształt "U"). + +- **Stopień 3 (np. `x³`)** → krzywa w kształcie "S". + + +**Jak rysować?** + +1. Znajdź **pierwiastki** (miejsca zerowe). + +2. Zobacz, czy wielomian **idzie w górę czy w dół** dla dużych `x`. + + +**Przykład:** +`W(x) = x² - 4` + +- Pierwiastki: `x = -2` i `x = 2`. + +- Dla `x → ∞` wielomian rośnie do `+∞`. + + +--- + +## **2.14. Nierówności wielomianowe** + +To gdy zamiast `=` jest `<`, `>`, `≤`, `≥`. + +**Jak rozwiązywać?** + +1. Rozłóż na czynniki. + +2. Znajdź pierwiastki. + +3. Narysuj **oś liczbową** i zaznacz przedziały. + + +**Przykład:** +`x² - 4 > 0` + +1. Rozkład: `(x - 2)(x + 2) > 0`. + +2. Pierwiastki: `x = -2`, `x = 2`. + +3. Rysujemy oś: + + ----(+)----(-2)----(-)----(2)----(+)----> + + - Dla `x < -2` → `(+) * (-) = (-)` → **nie spełnia**. + + - Dla `-2 < x < 2` → `(+) * (+) = (+)` → **spełnia**. + + - Dla `x > 2` → `(-) * (+) = (-)` → **nie spełnia**. + **Rozwiązanie:** `x ∈ (-∞, -2) ∪ (2, ∞)`.](<2.14. Nierówności wielomianowe +To gdy zamiast = jest %3C, %3E, ≤, ≥. + +Jak rozwiązywać? + +Rozłóż na czynniki. + +Znajdź pierwiastki. + +Narysuj oś liczbową i zaznacz przedziały. + +Przykład: +x² - 4 > 0 + +Rozkład: (x - 2)(x + 2) > 0. + +Pierwiastki: x = -2, x = 2. + +Rysujemy oś: + +----(+)----(-2)----(-)----(2)----(+)----> +Dla x < -2 → (+) * (-) = (-) → nie spełnia. + +Dla -2 < x < 2 → (+) * (+) = (+) → spełnia. + +Dla x > 2 → (-) * (+) = (-) → nie spełnia. +Rozwiązanie: x ∈ (-∞, -2) ∪ (2, ∞).>) \ No newline at end of file