NCERT Solutions for Class 9 Maths Exercise 2.4 Question 13
Understanding the Question 🧐
This question asks us to prove a special relationship that exists when the sum of three numbers is zero. We are given the condition that &&x+y+z=0&&, and based on this, we need to show that the sum of their cubes equals three times their product. This is a classic result derived from the main identity we’ve been using. This page provides a simple and clear proof as part of our comprehensive ncert solutions.
If &&x+y+z=0&&, show that &&x^3 + y^3 + z^3 = 3xyz&&.
Proof: Showing &&x^3 + y^3 + z^3 = 3xyz&& when &&x+y+z=0&& 📝
The proof for this statement is very straightforward and relies on the identity we learned in Question 11.
Step 1: State the relevant algebraic identity.
We know the identity involving three cubed variables:
&&x^3 + y^3 + z^3 – 3xyz = (x+y+z)(x^2 + y^2 + z^2 – xy – yz – zx)&&
Step 2: Apply the given condition to the identity.
The problem states that &&x+y+z=0&&. We can substitute this value into the right side of the identity.
&&x^3 + y^3 + z^3 – 3xyz = (0)(x^2 + y^2 + z^2 – xy – yz – zx)&&
Step 3: Simplify the Right Hand Side (RHS).
Any expression multiplied by zero results in zero. Therefore, the entire RHS of the equation becomes &&0&&.
&&x^3 + y^3 + z^3 – 3xyz = 0&&
Step 4: Rearrange the equation to achieve the desired result.
To get the final form, we can move the &&-3xyz&& term from the left side to the right side of the equation. Its sign will change from negative to positive.
&&x^3 + y^3 + z^3 = 3xyz&&
Conclusion:
We have successfully used the given condition to prove the required relationship.
Hence, it is shown that if &&x+y+z=0&&, then &&x^3 + y^3 + z^3 = 3xyz&&.
- This is a crucial conditional identity. It only works under the specific condition that the sum of the base terms is zero.
- This result provides a massive shortcut for calculations. Instead of cubing three numbers, you can just multiply their product by &&3&& (as you’ll see in the next question!).
- The proof relies entirely on the main identity: &&x^3 + y^3 + z^3 – 3xyz = (x+y+z)(…)&&.
FAQ
Q: What is the main algebraic identity used to start this proof?
A: The proof begins with the standard algebraic identity:
&&x^3 + y^3 + z^3 – 3xyz = (x+y+z)(x^2 + y^2 + z^2 – xy – yz – zx)&&.
Q: What is the specific condition given in this question?
A: The specific condition given is that the sum of the three variables is zero, i.e., &&x + y + z = 0&&.
Q: How does the condition &&x+y+z=0&& simplify the main identity?
A: When we substitute &&x+y+z=0&& into the identity, the entire right-hand side becomes &&0&& because anything multiplied by &&0&& is &&0&&. This simplifies the equation to &&x^3 + y^3 + z^3 – 3xyz = 0&&.
Q: What is the final relationship we are asked to prove?
A: We are asked to prove that the sum of the cubes of the three variables is equal to three times their product, i.e., &&x^3 + y^3 + z^3 = 3xyz&&.
Q: Why is this result useful?
A: This result is a very useful shortcut. If you need to find the sum of three cubes, you can first check if the sum of their base numbers is zero. If it is, you can find the answer by simply calculating &&3&& times their product, avoiding the difficult cubing process.
Further Reading
To learn more about conditional identities and their applications in algebra, you can refer to the official NCERT textbook on their website: https://ncert.nic.in/
