Search for: Let P(x) be a polynomial with real coefficients such that ∫01 xmP(1−x)dx=0∀m∈N∪{0}, then Let P(x) be a polynomial with real coefficients such that ∫01 xmP(1−x)dx=0∀m∈N∪{0}, then AP(x)=xn(1−x)n for some n∈NBP(x)=(1−x)2n for some n∈NCP(x)=1−xm(1−x)n for some m,n∈NDP(x) = 0 Fill Out the Form for Expert Academic Guidance!l Grade ---Class 1Class 2Class 3Class 4Class 5Class 6Class 7Class 8Class 9Class 10Class 11Class 12 Target Exam JEENEETCBSE +91 Preferred time slot for the call ---9 am10 am11 am12 pm1 pm2 pm3 pm4 pm5 pm6 pm7 pm8pm9 pm10pm Please indicate your interest Live ClassesBooksTest SeriesSelf Learning Language ---EnglishHindiMarathiTamilTeluguMalayalam Are you a Sri Chaitanya student? NoYes Verify OTP Code (required) I agree to the terms and conditions and privacy policy. Solution:0=∫01 xmP(1−x)dx=∫01 (1−x)mP(x)Let P(x)=∑k=0n akxk where ak∈R=∑k=0n ak[1−(1−x)]k=∑k=0n ak∑j=0k (−1)k kCj(1−x)j=∑k=0n bk(1−x)kwhere bk=∑j=kn (−1)k jCkajNow ∫01 (P(x))2dx=∫01 P(x)∑k=0n bk(1−x)kdx=∑k=0n bk∫01 (1−x)kP(x)dx=0As P(x) is a polynomial,P(x)≡0∀x∈[0,1]⇒ P(x)≡0 Related content How to Convert CGPA into Percentage CBSE Class 6 MCQs for All Subjects with Answers CBSE Class 9 English Rain on the Roof Worksheet CBSE MCQs on Crop Production and Management Class 8 with Answers CBSE MCQs On Mole Concept Class 11 With Answers CBSE Class 10 English Dust of Snow Worksheet CBSE Class 7 English Honeycomb Chapter 2 A Gift of Chappals MCQ Questions with Answers Top Career Option and Courses after 12th 2024: Arts, Commerce & Science Students CBSE Class 7 English MCQ Questions with Answers CBSE Class 12 English Lost Spring MCQ with Answers