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Q14(i):
Without actually calculating the cubes, find the value of each of the following:
(i) $(–12)^3 + (7)^3 + (5)^3$
Solution :
Step 1: Given Variables & Initial Setup
We are tasked with evaluating the sum of three cubes without performing direct cubing operations. The given mathematical expression is:
$(-12)^3 + (7)^3 + (5)^3$
Let us define the base of each cubic term as a distinct variable:
- $a = -12$
- $b = 7$
- $c = 5$
Step 2: Evaluating the Sum of the Base Variables
Before attempting to expand the expression, we must analyze the linear sum of the base variables. This is a critical diagnostic step in polynomial algebra to determine if a conditional identity applies.
$a + b + c = (-12) + 7 + 5$
$a + b + c = -12 + 12 = 0$
[Geometrically, this represents a closed vector loop in one dimension, where the displacement from the origin returns exactly to zero. See the precise vector visualization below.]
Step 3: Theoretical Justification & Algebraic Identity
We rely on the fundamental algebraic identity for the sum of three cubes:
$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$
[Per the Zero Product Property], if the first factor on the right-hand side, $(a + b + c)$, is equal to zero, the entire right-hand side of the equation evaluates to zero, regardless of the values of the quadratic polynomial factor. Therefore:
$a^3 + b^3 + c^3 - 3abc = (0) \times (a^2 + b^2 + c^2 - ab - bc - ca)$
$a^3 + b^3 + c^3 - 3abc = 0$
Transposing $-3abc$ to the right side yields the conditional identity:
If $a + b + c = 0$, then $a^3 + b^3 + c^3 = 3abc$.
Step 4: Application of the Conditional Identity
Since we have rigorously proven in Step 2 that $(-12) + 7 + 5 = 0$, we can directly substitute our variables into the conditional identity:
$(-12)^3 + (7)^3 + (5)^3 = 3 \cdot (-12) \cdot (7) \cdot (5)$
Step 5: Final Arithmetic Calculation
We now perform the sequential multiplication of the terms on the right-hand side:
- First, multiply the constants: $3 \times (-12) = -36$
- Next, multiply the remaining terms: $7 \times 5 = 35$
- Finally, compute the product of the two results: $(-36) \times 35$
To calculate $(-36) \times 35$ mentally or systematically:
$(-36) \times (30 + 5) = (-36 \times 30) + (-36 \times 5)$
$= -1080 - 180 = -1260$
Final Solution: The value of $(-12)^3 + (7)^3 + (5)^3$ is $-1260$.
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 $x^3 + y^3 + z^3 – 3xyz = \frac{1}{2}(x + y + z)[(x – y)^2 + (y – z)^2 + (z – x)^2]$
- Q13: If $x + y + z = 0$, show that $x^3 + y^3 + z^3 = 3xyz$.
- 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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