Take Tuition from the Best Tutors
Affordable fees 1-1 or Group class Flexible Timings Verified Tutors Search in
if x^2+y^2+z^2=xy+yz+zx then find the value of x^3+y^3+z^3
Asked by Sanju Last Modified
Answer 
We will work with the RHS part here,
(x + y+ z)( x² +y² +z² - xy - yz - zx)
Expanding the entire bracket,
= x³ + xy² + xz² - x²y - xyz -zx² + yx + y³ + yz² - xy² - y²z - xyz + zx² + zy² + z³ - xyz - yz² - z²x
=x³ + y³ +z³ - 3xyz
This is equal to LHS
Hence proved
read less
Given-
x^2+y^2+z^2=xy+yz+zx ---------(1)
multiplying by (x+y+z) on both sides-
x^2+y^2+z^2 (x+y+z)=(xy+yz+zx ) (x+y+z)
expand the above equations-
x^3+y^3+z^3+xy^2+x^2 y+yz^2+y^2 z+xz^2+x^2 z=x^2 y+xyz+zx^2+xy^2+
y^2 z+xyz+xyz+ yz^2+ xz^2
simplifying -
x^3+y^3+z^3=3xyz --Ans
As we know
(X^3 +Y^3+Z^3)-3XYZ = (X+Y+Z) (X^2 +Y^2 +Z^2-XY-YZ-ZX)
given
X^2 +Y^2 +Z^2=XY+YZ+ZX
I.e X^2 +Y^2 +Z^2-XY-YZ-ZX=0
so
(X^3 +Y^3+Z^3)-3XYZ = (X+Y+Z) (0)
(X^3 +Y^3+Z^3)-3XYZ=0
(X^3 +Y^3+Z^3)=3XYZ
So value of (X^3 +Y^3+Z^3) is (3XYZ)
read less
We will work with the RHS part here,
(x + y+ z)( x² +y² +z² - xy - yz - zx)
Expanding the entire bracket,
= x³ + xy² + xz² - x²y - xyz -zx² + yx + y³ + yz² - xy² - y²z - xyz + zx² + zy² + z³ - xyz - yz² - z²x
=x³ + y³ +z³ - 3xyz
This is equal to LHS
Hence proved
Harmanpreet Kaur
x^2+y^2+z^2=xy+yz+zx ----eq(1)
Identity is x^3+y^3+z^3 - 3xyz=(x+y+z)(x^2+y^2+z^2-xy-yz-zx)
x^3+y^3+z^3 -3xyz =(x+y+z)(xy+yz+zx-xy-yz-zx) (acc. to eq1.)
Therefore , x^3+y^3+z^3 - 3xyz = 0
So, x^3+y^3+z^3= 3xyz Answer
read less
Abhay Kumar
Math professional teacher with over 10 years of experience in teaching top institute of India
View 235 more Answers
Related Questions
Now ask question in any of the 1000+ Categories, and get Answers from Tutors and Trainers on UrbanPro.com
Ask a QuestionRelated Lessons
Recommended Articles
6 Exam Hall Tips To Follow, For Every Good Student
Appearing for exams could be stressful for students. Even though they might have prepared well, they could suffer from anxiety, tension etc. These are not good for their health and mind. However, following a few exam preparation tips can save them from all these and help them to score good marks. Let’s find out all...
Read full article >
How To Choose The Right School For Children?
Quality education does not only help children to get a successful career and life, but it also hugely contributes to society. The formal education of every child starts from school. Although there are numerous schools, parents find it challenging to choose the right one that would fit their child. It is difficult for them...
Read full article >
The Trend of E-learning and it’s Benefits
E-learning is not just about delivering lessons online. It has a much broader scope that goes beyond manual paper or PowerPoint Presentations. To understand the reach of E-learning and how the whole process works in developing the Educational system, we will discuss a few points here. Let us find out how this new learning...
Read full article >
6 Healthy Habits of School Children
While schools provide formal education to the children, the home is where they start learning about things informally. Parents think that schools will take the initiative to educate their children. Well, this is partially true, as parents also play an essential role in bringing up their child. For the development of particular...
Read full article >
Looking for Tuition ?
Learn from the Best Tutors on UrbanPro
Are you a Tutor or Training Institute?
Join UrbanPro Today to find students near youThe best tutors for Tuition Classes are on UrbanPro
The best Tutors for Tuition Classes are on UrbanPro
5
