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Eğitmen, \"ax² + 6x + 1 > 0\", \"ax² + 4x + 1 ≥ 0\", \"ax² + 6x - 3 \u003c 0\" gibi farklı türdeki eşitsizlikler üzerinden örnekler vererek çözüm tekniklerini göstermektedir."]},"endTime":3228,"title":"İkinci Dereceden Eşitsizlikler Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"İkinci Dereceden Eşitsizliklerin Tanıtımı","list":{"type":"unordered","items":["İkinci dereceden eşitsizlikler konusu dört derste anlatılacak ve önceki üç derste işaret tablosu anlatılmıştır.","İlk derste üç sayfalık ders yapılacak ve bu ders PDF olarak video açıklamasında paylaşılacaktır.","Konu, önce basit eşitsizlikler üzerinden ikinci dereceden eşitsizliklerin mantığı anlatılacaktır."]},"beginTime":3,"endTime":44,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=3&ask_summarization=1"},{"index":1,"title":"Basit Eşitsizliklerin Çözüm Yöntemleri","list":{"type":"unordered","items":["Basit eşitsizliklerin çözümü için iki yöntem kullanılabilir: basit eşitsizlik mantığı ve fonksiyon mantığı.","Basit eşitsizlik mantığıyla çözümde, bilinmeyeni yalnız bırakarak eşitsizliğin çözüm aralığı bulunur.","Fonksiyon mantığıyla çözümde, eşitsizliğin bir tarafı sıfır yapılır ve fonksiyonun işaret tablosu incelenerek çözüm aralığı belirlenir."]},"beginTime":44,"endTime":93,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=44&ask_summarization=1"},{"index":2,"title":"Fonksiyon Mantığıyla Basit Eşitsizlik Çözümü","list":{"type":"unordered","items":["Fonksiyon mantığıyla çözümde öncelikle fonksiyonun işaretsiz yapan noktaları bulunur.","İşaret tablosu yapılırken baş katsayının işareti son bölgedeki işaretini belirler ve tek katlı köklerde aralık değiştikçe işaret değişir.","Eşitsizlikte eşitlik varsa sıfır yapan noktalar da çözüm kümesine dahil edilir."]},"beginTime":93,"endTime":325,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=93&ask_summarization=1"},{"index":3,"title":"İkinci Dereceden Eşitsizliklerin Çözümü","list":{"type":"unordered","items":["İkinci dereceden eşitsizliklerde fonksiyonun işaretini inceleyerek çözüm kümesi bulunur.","Fonksiyonun işaretsiz yapan noktaları çarpanlarına ayırarak bulunur ve işaret tablosu yapılır.","Eşitsizlikte eşitlik varsa sıfır yapan noktalar çözüm kümesine dahil edilir, yoksa dahil edilmez."]},"beginTime":325,"endTime":443,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=325&ask_summarization=1"},{"index":4,"title":"Karmaşık İkinci Dereceden Eşitsizlikler","list":{"type":"unordered","items":["Karmaşık ikinci dereceden eşitsizliklerde fonksiyonun işaret tablosu yapılabilir.","Eşitsizlikte eşitlik yoksa sıfır yapan noktalar çözüm kümesine dahil edilmez.","Çözüm kümesi, tüm reel sayılar kümesinden belirli aralıklar çıkarılarak da gösterilebilir."]},"beginTime":443,"endTime":599,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=443&ask_summarization=1"},{"index":5,"title":"Eşitsizlik Çözümü Yöntemi","list":{"type":"unordered","items":["Eşitsizlik çözerken bir tarafında mutlaka sıfır olmalı, genellikle sıfır sağ tarafta tercih edilir.","Eşitsizliği çözmek için önce fonksiyonun işaret tablosu oluşturulur.","İşaret tablosu için önce fonksiyonun sıfıra eşit olduğu noktalar bulunur, sonra baş katsayıya bakılarak işaretler belirlenir."]},"beginTime":612,"endTime":612,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=612&ask_summarization=1"},{"index":6,"title":"İkinci Eşitsizlik Örneği","list":{"type":"unordered","items":["x² - 2x + 1 ≥ 0 eşitsizliği için x² - 2x + 1 = 0 denkleminin tek kökü x = 1'dir (çift katlı kök).","Çift katlı kök olduğu için işaret tablosunda işaret değişmez, fonksiyon her yerde pozitiftir.","Çözüm kümesi tüm reel sayılar olan R kümesidir."]},"beginTime":792,"endTime":902,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=792&ask_summarization=1"},{"index":7,"title":"Üçüncü Eşitsizlik Örneği","list":{"type":"unordered","items":["x² - 6x + 9 > 0 eşitsizliği için x² - 6x + 9 = 0 denkleminin tek kökü x = 3'tür (çift katlı kök).","Çift katlı kök olduğu için işaret tablosunda işaret değişmez, fonksiyon her yerde pozitiftir.","Çözüm kümesi (-∞, 3) ∪ (3, +∞) veya R - {3} olarak yazılabilir."]},"beginTime":902,"endTime":1022,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=902&ask_summarization=1"},{"index":8,"title":"Dördüncü Eşitsizlik Örneği","list":{"type":"unordered","items":["x² - 4x + 4 ≤ 0 eşitsizliği için x² - 4x + 4 = 0 denkleminin tek kökü x = 2'dir (çift katlı kök).","Çift katlı kök olduğu için işaret tablosunda işaret değişmez, fonksiyon her yerde pozitiftir.","Çözüm kümesi sadece x = 2 olan {2} kümesidir."]},"beginTime":1022,"endTime":1109,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1022&ask_summarization=1"},{"index":9,"title":"Beşinci Eşitsizlik Örneği","list":{"type":"unordered","items":["x² + 2x + 1 \u003c 0 eşitsizliği için x² + 2x + 1 = 0 denkleminin tek kökü x = -1'dir (çift katlı kök).","Çift katlı kök olduğu için işaret tablosunda işaret değişmez, fonksiyon her yerde pozitiftir.","Çözüm kümesi boş kümedir, çünkü fonksiyon hiçbir zaman sıfırdan küçük olmaz."]},"beginTime":1109,"endTime":1200,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1109&ask_summarization=1"},{"index":10,"title":"Altıncı Eşitsizlik Örneği","list":{"type":"unordered","items":["-x² + 2x - 1 \u003c 0 eşitsizliği için önce denklemin her iki tarafı -1 ile çarpılarak x² - 2x + 1 > 0 şeklinde yazılır.","x² - 2x + 1 = 0 denkleminin tek kökü x = 1'dir (çift katlı kök).","Çift katlı kök olduğu için işaret tablosunda işaret değişmez, fonksiyon her yerde pozitiftir."]},"beginTime":1200,"endTime":1275,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1200&ask_summarization=1"},{"index":11,"title":"İkinci Dereceden Eşitsizliklerin Çözümü","list":{"type":"unordered","items":["İkinci dereceden bir fonksiyonun işaret tablosu incelenerek çözüm kümesi bulunuyor.","Fonksiyonun baş katsayısı ve köklerinin katlılığı işaret değişimini belirliyor.","Çift katlı kök olduğu durumlarda fonksiyon işaret değiştirmiyor."]},"beginTime":1279,"endTime":1343,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1279&ask_summarization=1"},{"index":12,"title":"Örnek 1 - Eksi x²+6x-9\u003c0","list":{"type":"unordered","items":["Eksi x²+6x-9 ifadesi (x-3)² şeklinde çarpanlara ayrılıyor, bu da x=3 çift katlı kök olduğunu gösteriyor.","Fonksiyonun baş katsayısı eksi bir olduğu için fonksiyon her yerde negatif değer alıyor.","Çözüm kümesi R-{3} olarak bulunuyor, yani tüm reel sayılar kümesinden 3 çıkarılıyor."]},"beginTime":1343,"endTime":1469,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1343&ask_summarization=1"},{"index":13,"title":"Örnek 2 - Eksi x²+x-25≥0","list":{"type":"unordered","items":["Eksi x²+x-25 ifadesi (x-5)² şeklinde çarpanlara ayrılıyor, bu da x=5 çift katlı kök olduğunu gösteriyor.","Fonksiyonun baş katsayısı eksi bir olduğu için fonksiyon her yerde negatif değer alıyor.","Çözüm kümesi sadece {5} olarak bulunuyor, çünkü eşitsizlikte eşitlik durumu da dahil edilmiş."]},"beginTime":1469,"endTime":1577,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1469&ask_summarization=1"},{"index":14,"title":"Örnek 3 - Eksi x²+4x-4>0","list":{"type":"unordered","items":["Eksi x²+4x-4 ifadesi (x-2)² şeklinde çarpanlara ayrılıyor, bu da x=2 çift katlı kök olduğunu gösteriyor.","Fonksiyonun baş katsayısı eksi bir olduğu için fonksiyon her yerde negatif değer alıyor.","Çözüm kümesi boş küme olarak bulunuyor, çünkü fonksiyon hiçbir yerde sıfırdan büyük değil."]},"beginTime":1577,"endTime":1690,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1577&ask_summarization=1"},{"index":15,"title":"Örnek 4 - x²+2x+5≥0","list":{"type":"unordered","items":["x²+2x+5 ifadesinin delta değeri -16 olarak hesaplanıyor, bu da reel kök olmadığını gösteriyor.","Fonksiyonun baş katsayısı artı bir olduğu için fonksiyon her yerde pozitif değer alıyor.","Çözüm kümesi tüm reel sayılar kümesi olarak bulunuyor."]},"beginTime":1690,"endTime":1816,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1690&ask_summarization=1"},{"index":16,"title":"Örnek 5 - x²+x+1>0","list":{"type":"unordered","items":["x²+x+1 ifadesinin delta değeri -3 olarak hesaplanıyor, bu da reel kök olmadığını gösteriyor.","Fonksiyonun baş katsayısı artı bir olduğu için fonksiyon her yerde pozitif değer alıyor.","Çözüm kümesi tüm reel sayılar kümesi olarak bulunuyor."]},"beginTime":1816,"endTime":1894,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1816&ask_summarization=1"},{"index":17,"title":"İkinci Dereceden Eşitsizliklerin Çözümü","list":{"type":"unordered","items":["İkinci dereceden eşitsizliklerin çözümü için önce fonksiyonun işaret tablosu incelenir ve fonksiyonu sıfırdan küçük yapan noktalar tespit edilir.","Fonksiyonu sıfırdan küçük yapan noktaları bulmak için delta (discriminant) hesaplanır, delta negatif ise fonksiyon hiçbir noktada sıfıra eşit olmaz.","Fonksiyonun işaretini belirlemek için x² katsayısına bakılır; katsayı pozitifse fonksiyon pozitif değerler alır, negatifse negatif değerler alır."]},"beginTime":1898,"endTime":1910,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1898&ask_summarization=1"},{"index":18,"title":"Örnek Eşitsizliklerin Çözümü","list":{"type":"unordered","items":["x² - 2x + 3 \u003c 0 eşitsizliğinin çözüm kümesi boş kümedir çünkü delta negatif olup fonksiyon hiçbir noktada sıfıra eşit olmaz ve x² katsayısı pozitif olduğundan fonksiyon hep pozitiftir.","-x² + 2x - 4 > 0 eşitsizliğinin çözüm kümesi boş kümedir çünkü delta negatif olup fonksiyon hiçbir noktada sıfıra eşit olmaz ve x² katsayısı negatif olduğundan fonksiyon hep negatiftir.","-2x² + 3x - 4 > 0 eşitsizliğinin çözüm kümesi boş kümedir çünkü delta negatif olup fonksiyon hiçbir noktada sıfıra eşit olmaz ve x² katsayısı negatif olduğundan fonksiyon hep negatiftir."]},"beginTime":1910,"endTime":2387,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1910&ask_summarization=1"},{"index":19,"title":"Çözüm Kümesi Reel Sayılar Olan Eşitsizlikler","list":{"type":"unordered","items":["-3x² + 2x - 1 \u003c 0 eşitsizliğinin çözüm kümesi tüm reel sayılar kümesidir çünkü delta negatif olup fonksiyon hiçbir noktada sıfıra eşit olmaz ve x² katsayısı negatif olduğundan fonksiyon hep negatiftir.","x² + 2x - 7 \u003c 0 eşitsizliğinin çözüm kümesi tüm reel sayılar kümesidir çünkü delta negatif olup fonksiyon hiçbir noktada sıfıra eşit olmaz ve x² katsayısı negatif olduğundan fonksiyon hep negatiftir.","ax² + 6x + 1 > 0 eşitsizliğinin çözüm kümesi reel sayılar olduğuna göre, a'nın değer aralığı (9, +∞) olmalıdır."]},"beginTime":2387,"endTime":2617,"href":"/video/preview/17155920307412622592?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=2387&ask_summarization=1"},{"index":20,"title":"Eşitsizlik Çözümü ve Delta Analizi","list":{"type":"unordered","items":["ax² + 4x + 1 ≥ 0 eşitsizliğinin çözüm kümesi R ise, a'nın değer aralığı 4 ile +∞ arasındadır.","ax² + 6x - 3 \u003c 0 eşitsizliğinin çözüm kümesi R ise, a'nın değer aralığı (-∞, -3) arasındadır.","ax² + 3x - 1 ≤ 0 eşitsizliğinin çözüm kümesi R ise, a'nın değer aralığı (-∞, -9/4] 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arkadaşlarım merhabalar, yazılıda çıkabilecek sorulara hazırlandığımız bu derste Polinom konusunu tekrar edip üzerine bol bol soru çözeceğiz. Yazılınızdan yüksek notlar almanız dileğiyle...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/2814332/727099202a7215e811bcc20c89f19e60/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/NmGJRQIAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"3","reqid":"1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=videoid:7591879524079478624","teaser":[{"list":{"type":"unordered","items":["Bu video, matematik öğretmeni Yahya Özen tarafından sunulan, 10. sınıf öğrencilerine yönelik bir eğitim dersidir. Öğretmen, 2. dönem 1. yazılıya hazırlık amacıyla polinomlar konusunu detaylı şekilde anlatmaktadır.","Video, polinomların temel kavramlarından başlayarak (tanım, derece, baş katsayı, katsayılar toplamı, sabit terim) ileri seviye konulara (polinomların eşitliği, toplama, çıkarma, çarpma, bölme işlemleri, kalan bulma, çarpanlara ayırma) kadar uzanan kapsamlı bir içerik sunmaktadır. Öğretmen, konuları formüller yerine mantığını anlamayı vurgulayarak, ÖSYM benzeri sorular üzerinden konuyu pekiştirmektedir.","Videoda ayrıca çift ve tek dereceli terimlerin katsayılar toplamı formülleri (p(1) + p(-1) / 2 ve p(1) - p(-1) / 2), polinom bölmesi kuralları ve tam bölünme durumunda kalanın sıfır olması gibi önemli konular da ele alınmaktadır. Video, AYT matematik sınavına hazırlanan öğrenciler için de faydalı bilgiler içermektedir."]},"endTime":2895,"title":"10. Sınıf Polinomlar Konusu Eğitim Videosu","beginTime":0}],"fullResult":[{"index":0,"title":"10. Sınıf 2. Dönem 1. Yazılı Hazırlık","list":{"type":"unordered","items":["Matematik öğretmeni Yahya Özen, 10. sınıf 2. dönem 1. yazılıya hazırlık yapacağını belirtiyor.","Yazılı hazırlığı için polinom, çarpanlara ayırma ve denklemler gibi konular tekrar edilecek.","Hazırlık sonrası kolay, orta ve orta zor olmak üzere iki yazılı çözülecek."]},"beginTime":0,"endTime":32,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=0&ask_summarization=1"},{"index":1,"title":"Polinomların Tanımı","list":{"type":"unordered","items":["Polinom, değişkenlerin kuvveti doğal sayı olan fonksiyondur.","Değişkenler genellikle x ve y harfleri kullanılır, ancak z de kullanılabilir.","Polinomlar genellikle P(x), Q(x) ve R(x) şeklinde gösterilir."]},"beginTime":32,"endTime":103,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=32&ask_summarization=1"},{"index":2,"title":"Polinom Örnekleri","list":{"type":"unordered","items":["Polinomlarda sadece kuvvetlere odaklanılır, katsayılarla ilgili bir şey belirtilmez.","Kök x (x üzeri 1/2) ve x üzeri -2 gibi doğal sayı olmayan kuvvetler polinom belirtmez.","350x ifadesi üstel fonksiyon olduğu için polinom değildir."]},"beginTime":103,"endTime":214,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=103&ask_summarization=1"},{"index":3,"title":"Polinom Problemleri","list":{"type":"unordered","items":["Polinom belirtmesi için kuvvetin doğal sayı olması gerekir.","m değeri 5 ile 12 arasındaki doğal sayılar olabilir ve toplam 8 farklı değeri vardır.","n değeri 4'ten büyük eşit ve 32'nin tam böleni olmalıdır, bu durumda toplam 4 farklı değer vardır."]},"beginTime":214,"endTime":312,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=214&ask_summarization=1"},{"index":4,"title":"Polinomun Jargonları","list":{"type":"unordered","items":["Polinomun derecesi, en büyük kuvvettir.","Katsayılar, her bir terimin önündeki sayıdır.","Polinom terimi, artı ile eksi arasında kalan işareti ile beraber olan ifadedir."]},"beginTime":312,"endTime":374,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=312&ask_summarization=1"},{"index":5,"title":"Polinomların Derecesi ve Baş Katsayısı","list":{"type":"unordered","items":["Polinomun derecesi, en yüksek kuvveti olan terimdir ve köşeli parantez içinde gösterilir (der(px) = en yüksek kuvvet).","ÖSYM son yıllarda \"derecesi en yüksek olan ifadenin katsayısı\" veya \"en yüksek dereceli terimin katsayısı\" ifadelerini kullanmaktadır.","Sabit terim, x'in yanında olmayan ifadedir ve derecesi sıfırdır."]},"beginTime":376,"endTime":449,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=376&ask_summarization=1"},{"index":6,"title":"Polinom Değerleri Hesaplama","list":{"type":"unordered","items":["Polinomda belirli bir değer için hesaplama yaparken, x yerine o değeri koyarız.","Polinomda değişken değiştirme yaparken, x yerine istenen ifade yazılır.","Polinomları düzenlerken, x² teriminin en başta, x teriminin ortada ve sabitin sonunda olmasına dikkat edilmelidir."]},"beginTime":449,"endTime":524,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=449&ask_summarization=1"},{"index":7,"title":"Polinom Değerleri Örnekleri","list":{"type":"unordered","items":["Polinomda belirli bir değer için hesaplama yaparken, o değeri yerine koyarak sonucu buluruz.","Polinomda değişken değiştirme yaparken, verilen ifadeyi yerine yazarak yeni polinomu buluruz.","Polinomlarda verilen bilgileri kullanarak, istenen değerleri hesaplayabiliriz."]},"beginTime":524,"endTime":691,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=524&ask_summarization=1"},{"index":8,"title":"Polinomlarda Katsayılar ve Sabit Terim Bulma","list":{"type":"unordered","items":["Polinomda katsayılar toplamı bulmak için x yerine 1 yazılır.","Sabit terim bulmak için x yerine 0 şeklinde yazılır.","Farklı polinomlarda katsayılar toplamı bulmak için x yerine 1 yazılır, sabit terim bulmak için x yerine 0 şeklinde yazılır."]},"beginTime":695,"endTime":738,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=695&ask_summarization=1"},{"index":9,"title":"Polinom Problemleri Çözümü","list":{"type":"unordered","items":["P(x) polinomunun katsayılar toplamı P(1) olarak hesaplanır.","Sabit terim bulmak için x yerine 0 şeklinde yazılır.","Farklı polinomlarda katsayılar toplamı ve sabit terim bulmak için x yerine uygun değerler verilir."]},"beginTime":738,"endTime":900,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=738&ask_summarization=1"},{"index":10,"title":"Polinomlarda Özel Durumlar","list":{"type":"unordered","items":["Sabit terim bulmak için x yerine 0 şeklinde yazılır.","Polinomlarda katsayılar toplamı bulmak için x yerine 1 yazılır.","Çift dereceli terimlerin katsayılar toplamı P(1) + P(-1) / 2, tek dereceli terimlerin katsayılar toplamı P(1) - P(-1) / 2 formülüyle hesaplanır."]},"beginTime":900,"endTime":1190,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=900&ask_summarization=1"},{"index":11,"title":"Sabit Polinomlar","list":{"type":"unordered","items":["Sabit fonksiyon içinde x bulunmayan ifadedir.","Sabit polinom içinde x bulunmayan polinomdur.","Özel polinom sıfıra eşitse polinom olarak adlandırılır."]},"beginTime":1190,"endTime":1221,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1190&ask_summarization=1"},{"index":12,"title":"Polinomların Eşitliği","list":{"type":"unordered","items":["P(x) bir polinom, Q(x) sabit bir polinom olduğunda P(x) + Q(x) = P(x) + a şeklinde yazılır.","Polinomlarda eşitlik durumunda x yerine a yazarak denklem çözülebilir.","Denklemin çözümü için çarpanlarına ayırma yöntemi kullanılabilir."]},"beginTime":1221,"endTime":1311,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1221&ask_summarization=1"},{"index":13,"title":"Polinomların Eşitliği","list":{"type":"unordered","items":["İki polinomun eşitliği verildiğinde, dereceleri aynı olan ifadelerin katsayıları da aynı olmalıdır.","Tek dereceli terimlerin katsayıları toplamı formülü P(1) - P(-1) / 2 kullanılarak hesaplanabilir.","Polinomlarda toplama ve çıkarma yaparken, dereceleri aynı olan terimlerin katsayıları toplanır veya çıkarılır."]},"beginTime":1322,"endTime":1529,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1322&ask_summarization=1"},{"index":14,"title":"Polinomlarda Çarpma İşlemi","list":{"type":"unordered","items":["Polinomlarda çarpma işlemi sorularında, belirli dereceli terimlerin katsayıları sorulabilir.","Belirli dereceli terimlerin katsayılarını bulmak için, en yüksek dereceli terimlerin çarpımları incelenir.","Çarpma işleminde, x² ile x² çarpıldığında x⁴, x ile x³ çarpıldığında x⁴ gibi terimler elde edilir."]},"beginTime":1529,"endTime":1633,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1529&ask_summarization=1"},{"index":15,"title":"Polinomlarda Bölme İşlemi","list":{"type":"unordered","items":["Polinomlarda bölme işleminde, kalanın derecesi bölenin derecesinden mutlaka küçük olmalıdır.","Polinom diğer polinoma tam bölünüyorsa, kalan sıfırdır ve bölen polinom çarpan olarak bulunur.","Polinom bölmesinde, bölen ile bölümün çarpımı artı kalan formülü kullanılır."]},"beginTime":1633,"endTime":1770,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1633&ask_summarization=1"},{"index":16,"title":"Polinom Değerleri","list":{"type":"unordered","items":["Polinom değerleri hesaplanırken, x yerine belirli değerler yazılabilir.","Polinom bölmesi yaparak, bilinmeyen polinom ifadeleri bulunabilir.","Polinom bölmesinde, ortak çarpan parantezine alma yöntemi kullanılabilir."]},"beginTime":1770,"endTime":1966,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1770&ask_summarization=1"},{"index":17,"title":"Polinomların Derecesi","list":{"type":"unordered","items":["Polinomun derecesi, içindeki en yüksek kuvveti ifade eder.","Dereceyi hesaplamak için formülleri ezberlemek yerine, polinomun içindeki en yüksek kuvveti bulmak yeterlidir.","Polinomun derecesini etkileyen sadece en yüksek kuvvettir, katsayılar ve sabit terimler dereceyi etkilemez."]},"beginTime":1976,"endTime":2017,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=1976&ask_summarization=1"},{"index":18,"title":"Polinomların Derecesi Örnekleri","list":{"type":"unordered","items":["P(x) = x³ ve Q(x) = x² olduğunda, P(2x+4) ifadesinin derecesi 3'tür çünkü 2x+4'ün küpü alındığında 3'ten büyük bir kuvvet çıkmaz.","P(x) = x³ + 4x² + 4x + 1 ve Q(x) = x² olduğunda, P(x)Q(x) ifadesinin derecesi 5'tir çünkü x³ ile x²'nin çarpımı x⁵'tir.","P(x) = x³ ve Q(x) = x² olduğunda, P(x)Q(x) / x² ifadesinin derecesi 1'dir çünkü tabanlar aynı olduğunda kuvvetler çıkarılır."]},"beginTime":2017,"endTime":2205,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=2017&ask_summarization=1"},{"index":19,"title":"Karmaşık Polinom Örnekleri","list":{"type":"unordered","items":["P(x) = x²ᵃ ve Q(x) = xᵇ olduğunda, P(x)²Q(x) ifadesinin derecesi 8 olduğuna göre 2a + b = 8 bulunur.","P(x)³Q(x) / x² = x² olduğuna göre 3a - b = 2 bulunur ve bu iki denklem taraf tarafa toplanarak a = 2 ve b = 4 bulunur.","P(x) = x² + 3x + 5 ve Q(x) = x⁴ olduğunda, P(x)²Q(x) ifadesinin derecesi 8'dir çünkü x²'nin karesi ile x⁴'ün çarpımı x⁸'tir."]},"beginTime":2205,"endTime":2346,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=2205&ask_summarization=1"},{"index":20,"title":"Polinomlarda Kalan Bulma","list":{"type":"unordered","items":["Polinomlarda ax + b şeklindeki birinci derece ifadenin kalanını bulmak için x = -b/a değerini polinoma yerleştiririz.","P(x) polinomunun x-2 ile bölümünden kalanı bulmak için x = 2 değerini polinoma yerleştiririz.","P(2x-3) polinomunun x+5 ile bölümünden kalanı bulmak için önce x+5 = 0 denkleminden x = -5 bulunur, sonra 2x-3 = -5 denkleminden x = -1 bulunur ve P(-1) hesaplanır."]},"beginTime":2346,"endTime":2545,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=2346&ask_summarization=1"},{"index":21,"title":"Polinomlarda Bölme ve Çarpanlar","list":{"type":"unordered","items":["Polinomlarda bölme işlemi yapılırken, tam bölünüyorsa kalan sıfırdır ve bölünen polinomun bölen polinomun bir çarpanı olduğu belirtilir.","Bir polinomun x-3 ile tam bölünmesi durumunda, polinomun içerisinde kesinlikle x-3 polinomu vardır.","ÖSYM sınavlarında bu tür soruların çok benzerleri sorulabilir ve yazılarda mutlaka 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yazılabilir ve P(1)=-20 olarak bulunur.","Polinomların örüntüleri olmayabilir, bu durumda her kök için ayrı hesaplama yapılmalıdır."]},"beginTime":2686,"endTime":2789,"href":"/video/preview/7591879524079478624?parent-reqid=1769628201203356-7088343846293542710-balancer-l7leveler-kubr-yp-sas-99-BAL&text=Math-Tech+with+YAHYA&t=2686&ask_summarization=1"},{"index":24,"title":"Kareli Polinom Problemi","list":{"type":"unordered","items":["P(1)=1, P(2)=4, P(3)=9 verildiğinde, polinomun katsayıları hesaplanarak P(x)=x³-2x²-4x-3+x² olarak bulunur.","P(4) değeri hesaplanarak 22 olarak bulunur.","Yeni ders çarpanlara ayırma konusuna 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