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Q13:
If $x + y + z = 0$, show that $x^3 + y^3 + z^3 = 3xyz$.
Solution :
Given Variables & Initial Setup
We are given the linear equation involving three variables:
$x + y + z = 0$
Objective: Prove the algebraic relationship $x^3 + y^3 + z^3 = 3xyz$.
To provide a comprehensive, masterclass-level proof, this relationship will be demonstrated using two distinct mathematical approaches: the Standard Identity Method and the Direct Algebraic Manipulation (Cubing) Method.
Method 1: Utilizing the Standard Algebraic Identity
Step 1: State the relevant polynomial identity
In algebra, the sum of three cubes is governed by the following fundamental identity:
$x^3 + y^3 + z^3 - 3xyz = (x + y + z)(x^2 + y^2 + z^2 - xy - yz - zx)$
[Theoretical Justification: This identity is derived by expanding the right-hand side and canceling out the intermediate cross-terms, leaving only the sum of the cubes and the $-3xyz$ term.]
Step 2: Substitute the given condition
We are given the premise that $x + y + z = 0$. We substitute this value directly into the right-hand side of our identity:
$x^3 + y^3 + z^3 - 3xyz = (0) \cdot (x^2 + y^2 + z^2 - xy - yz - zx)$
Step 3: Apply the Zero Product Property
According to the Zero Product Property, any finite real number or algebraic expression multiplied by zero results in zero. Therefore, the entire right-hand side collapses to $0$:
$x^3 + y^3 + z^3 - 3xyz = 0$
Step 4: Isolate the sum of the cubes
By adding $3xyz$ to both sides of the equation, we arrive at the final required expression:
$x^3 + y^3 + z^3 = 3xyz$
Method 2: Direct Algebraic Manipulation (Cubing)
Step 1: Rearrange the initial equation
Starting with the given condition, isolate two variables on one side of the equation:
$x + y + z = 0 \implies x + y = -z$
Step 2: Cube both sides of the equation
To generate the cubic terms required for the proof, apply the power of 3 to both sides:
$(x + y)^3 = (-z)^3$
Step 3: Expand using the binomial cube identity
Expand the left side using the standard binomial identity $(a + b)^3 = a^3 + b^3 + 3ab(a + b)$. Note that the cube of a negative value remains negative, so $(-z)^3 = -z^3$.
$x^3 + y^3 + 3xy(x + y) = -z^3$
Step 4: Substitute the initial rearranged condition
From Step 1, we established that $(x + y) = -z$. Substitute $-z$ back into the expanded equation in place of $(x + y)$:
$x^3 + y^3 + 3xy(-z) = -z^3$
Step 5: Simplify and rearrange terms
Multiply the terms to simplify the equation:
$x^3 + y^3 - 3xyz = -z^3$
Finally, transpose $-z^3$ to the left side (becoming $+z^3$) and $-3xyz$ to the right side (becoming $+3xyz$):
$x^3 + y^3 + z^3 = 3xyz$
Visual Representation of the Logical Pathways
The following flowchart illustrates the dual algebraic pathways utilized to prove the theorem, confirming the structural integrity of both methods.
Final Conclusion
Both the application of the standard cubic polynomial identity and direct algebraic expansion yield the exact same mathematical truth. When the sum of three variables is zero, the sum of their cubes is perfectly balanced by three times their product.
Final Solution: It is proven that if $x + y + z = 0$, then $x^3 + y^3 + z^3 = 3xyz$.
More Questions from Class 9 Mathematics Polynomials EXERCISE 2.4
- Q1(i): Use suitable identities to find the following products: (i) $(x + 4) (x + 10)$
- Q1(ii): Use suitable identities to find the following products: (ii) $(x + 8) (x – 10)$
- Q1(iii): Use suitable identities to find the following products: (iii) $(3x + 4) (3x – 5)$
- Q1(iv): Use suitable identities to find the following products: (iv) $(y^2 + \frac{3}{2}) (y^2 – \frac{3}{2})$
- Q1(v): Use suitable identities to find the following products: (v) $(3 – 2x) (3 + 2x)$
- Q10(i): Factorise each of the following: (i) $27y^3 + 125z^3$ [Hint : See Question 9.]
- Q10(ii): Factorise each of the following: (ii) $64m^3 – 343n^3$ [Hint : See Question 9.]
- Q11: Factorise : $27x^3 + y^3 + z^3 – 9xyz$
- Q12: Verify that $x^3 + y^3 + z^3 – 3xyz = \frac{1}{2}(x + y + z)[(x – y)^2 + (y – z)^2 + (z – x)^2]$
- Q14(i): Without actually calculating the cubes, find the value of each of the following: (i) $(–12)^3 + (7)^3 + (5)^3$
- Q14(ii): Without actually calculating the cubes, find the value of each of the following: (ii) $(28)^3 + (–15)^3 + (–13)^3$
- Q15(i): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (i) Area : $25a^2 – 35a + 12$
- Q15(ii): Give possible expressions for the length and breadth of each of the following rectangles, in which their areas are given: (ii) Area : $35y^2 + 13y –12$
- Q16(i): What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (i) Volume : $3x^2 – 12x$
- Q16(ii): What are the possible expressions for the dimensions of the cuboids whose volumes are given below? (ii) Volume : $12ky^2 + 8ky – 20k$
- Q2(i): Evaluate the following products without multiplying directly: (i) $103 \times 107$
- Q2(ii): Evaluate the following products without multiplying directly: (ii) $95 \times 96$
- Q2(iii): Evaluate the following products without multiplying directly: (iii) $104 \times 96$
- Q3(i): Factorise the following using appropriate identities: (i) $9x^2 + 6xy + y^2$
- Q3(ii): Factorise the following using appropriate identities: (ii) $4y^2 – 4y + 1$
- Q3(iii): Factorise the following using appropriate identities: (iii) $x^2 – \frac{y^2}{100}$
- Q4(i): Expand each of the following, using suitable identities: (i) $(x + 2y + 4z)^2$
- Q4(ii): Expand each of the following, using suitable identities: (ii) $(2x – y + z)^2$
- Q4(iii): Expand each of the following, using suitable identities: (iii) $(–2x + 3y + 2z)^2$
- Q4(iv): Expand each of the following, using suitable identities: (iv) $(3a – 7b – c)^2$
- Q4(v): Expand each of the following, using suitable identities: (v) $(–2x + 5y – 3z)^2$
- Q4(vi): Expand each of the following, using suitable identities: (vi) $(\frac{1}{4}a - \frac{1}{2}b + 1)^2$
- Q5(i): Factorise: (i) $4x^2 + 9y^2 + 16z^2 + 12xy – 24yz – 16xz$
- Q5(ii): Factorise: (ii) $2x^2 + y^2 + 8z^2 – 2\sqrt{2}xy + 4\sqrt{2}yz – 8xz$
- Q6(i): Write the following cubes in expanded form: (i) $(2x + 1)^3$
- Q6(ii): Write the following cubes in expanded form: (ii) $(2a – 3b)^3$
- Q6(iii): Write the following cubes in expanded form: (iii) $(\frac{3}{2}x + 1)^3$
- Q6(iv): Write the following cubes in expanded form: (iv) $(x - \frac{2}{3}y)^3$
- Q7(i): Evaluate the following using suitable identities: (i) $(99)^3$
- Q7(ii): Evaluate the following using suitable identities: (ii) $(102)^3$
- Q7(iii): Evaluate the following using suitable identities: (iii) $(998)^3$
- Q8(i): Factorise each of the following: (i) $8a^3 + b^3 + 12a^2b + 6ab^2$
- Q8(ii): Factorise each of the following: (ii) $8a^3 – b^3 – 12a^2b + 6ab^2$
- Q8(iii): Factorise each of the following: (iii) $27 – 125a^3 – 135a + 225a^2$
- Q8(iv): Factorise each of the following: (iv) $64a^3 – 27b^3 – 144a^2b + 108ab^2$
- Q8(v): Factorise each of the following: (v) $27p^3 – \frac{1}{216} – \frac{9}{2}p^2 + \frac{1}{4}p$
- Q9(i): Verify : (i) $x^3 + y^3 = (x + y) (x^2 – xy + y^2)$
- Q9(ii): Verify : (ii) $x^3 – y^3 = (x – y) (x^2 + xy + y^2)$
CBSE Solutions for Class 9 Mathematics Polynomials
Chapters in CBSE - Class 9 Mathematics
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