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Q10(ii):
Factorise each of the following:
(ii) $64m^3 – 343n^3$
[Hint : See Question 9.]
Solution :
Given Expression & Initial Setup
We are tasked with factorising the following algebraic expression:
$64m^3 - 343n^3$
Upon initial inspection, the expression consists of two terms separated by a minus sign. The variables $m$ and $n$ are both raised to the third power, which strongly indicates that we must evaluate the numerical coefficients to determine if they are also perfect cubes.
Step 1: Identifying Perfect Cubes
To apply the relevant algebraic identity, we must rewrite both terms entirely as perfect cubes in the form of $x^3$ and $y^3$.
- First Term ($64m^3$): The prime factorization of $64$ is $2^6$, which can be grouped as $(2^2)^3 = 4^3$. Therefore, the entire first term can be rewritten as:
$64m^3 = (4m)^3$ - Second Term ($343n^3$): The prime factorization of $343$ is $7 \times 7 \times 7 = 7^3$. Therefore, the entire second term can be rewritten as:
$343n^3 = (7n)^3$
Substituting these back into the original expression yields:
$(4m)^3 - (7n)^3$
Step 2: Stating the Relevant Algebraic Identity
The expression is now explicitly in the form of a difference of two cubes. [Per the fundamental algebraic identities of polynomials], the difference of two cubes is factored using the following formula:
$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$
Step 3: Substitution and Expansion
By mapping our specific terms to the identity, we establish the following equivalencies:
- $x = 4m$
- $y = 7n$
We now substitute these values directly into the right-hand side of the identity, $(x - y)(x^2 + xy + y^2)$:
$= (4m - 7n) \left[ (4m)^2 + (4m)(7n) + (7n)^2 \right]$
Step 4: Final Simplification
To achieve the final factorised form, we must expand the terms within the square brackets by applying the exponent rules [specifically $(ab)^n = a^n b^n$] and performing basic multiplication:
- Square the first term: $(4m)^2 = 4^2 \cdot m^2 = 16m^2$
- Multiply the two terms: $(4m)(7n) = 4 \cdot 7 \cdot m \cdot n = 28mn$
- Square the second term: $(7n)^2 = 7^2 \cdot n^2 = 49n^2$
Substituting these simplified components back into our factored expression yields:
$= (4m - 7n)(16m^2 + 28mn + 49n^2)$
Final Solution: $(4m - 7n)(16m^2 + 28mn + 49n^2)$
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.]
- 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(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) $x^3 – y^3 = (x – y) (x^2 + xy + y^2)$
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