UrbanPro

Location Set your Location

Login Ask & Answer Success Stories Paid Courses Free Classes Tuition & Classes Fees Write a Review All Categories Help Center
Post your Learning Need Signup as a Tutor
✕

Popular Cities

Bangalore

Hyderabad

Delhi

Mumbai

Chennai

Pune

Kolkata

Gurgaon

Ahmedabad

Jaipur

Noida

✕

Take Class 10 Tuition from the Best Tutors

  • Affordable fees
  • 1-1 or Group class
  • Flexible Timings
  • Verified Tutors

Book a Free Demo

Class 10 > Learn Mathematics > If the zeroes of polynomial x3 -- ax2 + bx -- c are...
✕

Search in

If the zeroes of polynomial x3 – ax2 + bx – c are in AP then show that 2a3 – 9ab + 27c = 0

Asked by Last Modified  

5 Answers

Learn Mathematics +1 Class IX-X Tuition

Follow 4
Answer

Please enter your answer

Mathematics

zreors of polynomial means roots of polynomial if the roots of polynomial are in AP then the roots are A+d, A, A-d Sum of the roots = (-) Co-efficient of X^2/ Co-efficient of X^3 = A+d +A +A-d -(-a) = 3A a =3A A= a/3 since...
read more
zreors of polynomial means roots of polynomial if the roots of polynomial are in AP then the roots are A+d, A, A-d Sum of the roots = (-) Co-efficient of X^2/ Co-efficient of X^3 = A+d +A +A-d -(-a) = 3A a =3A A= a/3 since f(x)= x^3 - ax^2 +bx - c f(A)= A^3 - aA^2 +bA -c put A= a/3 (a/3)^3 -a(a/3)^2 +b(a/3) -c=0 (a^3)/27 - (a^3)/9 +ab/3 -c = 0 (a^3) -3(a^3) +9ab -27c =0 -2(a^3) +9ab - 27c = 0 2a^3 -9ab +27c = 0 hence PROVED read less
Comments
Dislike Bookmark

let l , m , n are zeroes of given polynomial. now sum of l+m+n = a product lmn = c product of zeros taken two at time lm+mn+nl = b also it is given that zeros of polynomial are in A. P . then 2m= l+ n from these equations we get a = 3m , b=2m2 + ln. now putting values...
read more
let l , m , n are zeroes of given polynomial. now sum of l+m+n = a product lmn = c product of zeros taken two at time lm+mn+nl = b also it is given that zeros of polynomial are in A. P . then 2m= l+ n from these equations we get a = 3m , b=2m2 + ln. now putting values of a, b , c in LHL we get 0. read less
Comments
Dislike Bookmark

Tutor

Let p-1, p, p+1 be the zeroes of the given polynomial. p-1+p+p+1=a 3p=a (p-1)p+p(p+1)+(p-1)(p+1)=b p2-p+p2+p+p2-1=b 3p2-1=b (p-1)p(p+1)=c (p2-1)p=c p3-p=c Consider 2a3-9ab+27c 2(3p)3-9(3p)(3p2-1)+27(p3-p)= 54p3-81p3+27p+27p3-27p= 81p3-81p3=0 Therefore 2a3-9ab+27c=0
Comments
Dislike Bookmark

Trainer

Let the zeros of given polynomial is m-d, m, and m+d. Here m and d are not zero. substituting the value of zeros in the polynomial, we have=>(m-d)^3-a(m-d)^2+b(m-d)-c=0 =>m^3-d^3-3m^2b+3md^2-a*(m^2+d^2-2*m*d)+b*m-b*d-c=0 =>m^3-d^3-3m^2b+3md^2-a*m^2-ad^2+2*a*m*d+b*m-b*d-c=0----------------(i) Similarily,...
read more
Let the zeros of given polynomial is m-d, m, and m+d. Here m and d are not zero. substituting the value of zeros in the polynomial, we have=>(m-d)^3-a(m-d)^2+b(m-d)-c=0 =>m^3-d^3-3m^2b+3md^2-a*(m^2+d^2-2*m*d)+b*m-b*d-c=0 =>m^3-d^3-3m^2b+3md^2-a*m^2-ad^2+2*a*m*d+b*m-b*d-c=0----------------(i) Similarily, we get a new equation by putting the (m+d), =>m^3+d^3+3m*d^2-a*m^2-a*d*d^2-2a*m*d+b*m+b*d-c=0---------------------(ii) Again, Putting the value of m in polynomial, we have m^3-a*m^2+b*m-c=0-----------------------------------------(iii) Adding equation (i) & (ii), we have=>2*m^3+6*m*d^2-2a*m^2-2a*d^2+2b*m-2*c=0 =>2*(3^3-a*m^2+b*m-c)+6m*d^2-2*a*d^2=0 =>6*m*d^2=2*a*d^2 =>m=a/3------------------------------(iv) Now, Putting the value of m in the polynomial, =>(a/3)^3-a*(a/3)^2+b*a/3-c=0 =>a^3/27-a^3/9+a*b/3-c=0 =>a^3-3*a^3+9*a*b-27*c=0 =>2*a^3-9*a*b+27*c=0 (Proved) read less
Comments
Dislike Bookmark

As we know, sum of roots = -(coeff. of x2)/(coeff. of x3) = a Since the roots ( or zeros) of polynomial are in AP, then we can assume the roots to be A-D, A, A+D ( with D as common diff.) sum of roots = A-D+A+A+D = 3A =a therefore A = a/3 Since, A is a zero of the polynomial, then it should satisfy...
read more
As we know, sum of roots = -(coeff. of x2)/(coeff. of x3) = a Since the roots ( or zeros) of polynomial are in AP, then we can assume the roots to be A-D, A, A+D ( with D as common diff.) sum of roots = A-D+A+A+D = 3A =a therefore A = a/3 Since, A is a zero of the polynomial, then it should satisfy the eqn. put A=a/3 in eqn (a^3)/27 - (a^3)/9 + (a*b)/3 - c = 0 Hence, 2a3-9ab+27c = 0 read less
Comments
Dislike Bookmark

View 3 more Answers

Related Questions

What is Pythagoras theorem?
Sum of the squares of adjacent side and opposite sides will be equal to square of the hypotenuse in a right angled triangle.
Kudo
Simplify: log base 6 (216) + [log (42) - log (6)] / log (49)?
216 = 6^3; log 42 - log 6 = log (42/6) = log 7 ; / log (49) = log 7 / 2 log 7 = 1/2 ; Answer = 3 + 1/2 = 3.5
Vishal
0 0
5
What is the best all day timetable for class 9th & 10th students?
It depends on syllabus...n also depends on student's ability
Sriomprakash
0 0
5
Is it necessary to pay for computer tuition in class 10 ICSE?
YES
Ushashree
0 0
5
One side of a rectangle exceeds its other side by 2 cm. If its area is 195 sq. cm, determine the sides of the rectangle?
13, 15.
Vikram
0 0
5

Now ask question in any of the 1000+ Categories, and get Answers from Tutors and Trainers on UrbanPro.com

Ask a Question

Recommended Articles

Meet Raghunandan.G.H, a B. Tech Tutor from...

Raghunandan is a passionate teacher with a decade of teaching experience. Being a skilled trainer with extensive knowledge, he provides high-quality BTech, Class 10 and Class 12 tuition classes. His methods of teaching with real-time examples makes difficult topics simple to understand. He explains every concept in-detail...

Read full article >

Meet Mohammad Wazid, a skilled trainer for...

Mohammad Wazid is a certified professional tutor for class 11 students. He has 6 years of teaching experience which he couples with an energetic attitude and a vision of making any subject easy for the students. Over the years he has developed skills with a capability of understanding the requirements of the students. This...

Read full article >

Meet Sandhya R, a B.Sc tutor from Bangalore

Sandhya is a proactive educationalist. She conducts classes for CBSE, PUC, ICSE, I.B. and IGCSE. Having a 6-year experience in teaching, she connects with her students and provides tutoring as per their understanding. She mentors her students personally and strives them to achieve their goals with ease. Being an enthusiastic...

Read full article >

Quest Academy - Institute of the month

Quest Academy is a professional Bangalore based NEET and JEE (Main + Advanced) training institute. The academy was incorporated in 2015 to cater to the needs of students, who aim to crack competitive exams by connecting with the best brains around. The institute helps students enhance their skills and capabilities through...

Read full article >

Find Grade 10 Tuition near you
Grade 10 Tuition in Delhi
Grade 10 Tuition in Bangalore
Grade 10 Tuition in Hyderabad
Grade 10 Tuition in Kolkata
Grade 10 Tuition in Mumbai
Grade 10 Tuition in Chennai
Grade 10 Tuition in Pune
Grade 10 Tuition in Noida
Grade 10 Tuition in Lucknow
Grade 10 Tuition in Jaipur

Looking for Class 10 Tuition ?

Learn from the Best Tutors on UrbanPro

Are you a Tutor or Training Institute?

Join UrbanPro Today to find students near you
X

Looking for Class 10 Tuition Classes?

The best tutors for Class 10 Tuition Classes are on UrbanPro

  • Select the best Tutor
  • Book & Attend a Free Demo
  • Pay and start Learning

Take Class 10 Tuition with the Best Tutors

The best Tutors for Class 10 Tuition Classes are on UrbanPro

This website uses cookies

We use cookies to improve user experience. Choose what cookies you allow us to use. You can read more about our Cookie Policy in our Privacy Policy

Accept All
Decline All

UrbanPro.com is India's largest network of most trusted tutors and institutes. Over 55 lakh students rely on UrbanPro.com, to fulfill their learning requirements across 1,000+ categories. Using UrbanPro.com, parents, and students can compare multiple Tutors and Institutes and choose the one that best suits their requirements. More than 7.5 lakh verified Tutors and Institutes are helping millions of students every day and growing their tutoring business on UrbanPro.com. Whether you are looking for a tutor to learn mathematics, a German language trainer to brush up your German language skills or an institute to upgrade your IT skills, we have got the best selection of Tutors and Training Institutes for you. Read more

ThinkVidya Learning Pvt Ltd © 2010-2024All Rights Reserved