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Q8(iv):
Factorise each of the following:
(iv) $64a^3 – 27b^3 – 144a^2b + 108ab^2$
Solution :
Step 1: Initial Setup and Polynomial Rearrangement
We are given the following polynomial expression of degree 3:
$P(a,b) = 64a^3 - 27b^3 - 144a^2b + 108ab^2$
To facilitate the identification of standard algebraic structures, we rearrange the terms in descending order of the powers of the first variable, $a$, and ascending order of the powers of the second variable, $b$.
$P(a,b) = 64a^3 - 144a^2b + 108ab^2 - 27b^3$
Step 2: Identification of the Relevant Algebraic Identity
The rearranged polynomial consists of four terms, with alternating signs ($+, -, +, -$), and the first and last terms appear to be perfect cubes. This structural signature strongly indicates the application of the standard binomial expansion identity for the cube of a difference [Per the Binomial Theorem for $(x-y)^n$ where $n=3$]:
$(x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3$
Step 3: Extracting the Base Terms ($x$ and $y$)
We equate the first and last terms of our polynomial to the corresponding terms in the identity to find the base variables $x$ and $y$.
- First Term ($x^3$):
$x^3 = 64a^3$
Taking the principal cube root of both sides:
$x = \sqrt[3]{64a^3} = \sqrt[3]{4^3 \cdot a^3} = 4a$ - Last Term ($y^3$):
$y^3 = 27b^3$
Taking the principal cube root of both sides:
$y = \sqrt[3]{27b^3} = \sqrt[3]{3^3 \cdot b^3} = 3b$
Step 4: Verification of the Cross Terms
To rigorously prove that the polynomial is indeed a perfect cube, we must verify that the middle terms of the given expression exactly match the $-3x^2y$ and $+3xy^2$ terms of the identity when $x = 4a$ and $y = 3b$.
- Verifying the second term ($-3x^2y$):
$-3x^2y = -3(4a)^2(3b)$
$= -3(16a^2)(3b)$
$= -48a^2(3b) = -144a^2b$
[This perfectly matches the second term of our rearranged polynomial.] - Verifying the third term ($+3xy^2$):
$+3xy^2 = 3(4a)(3b)^2$
$= 3(4a)(9b^2)$
$= 12a(9b^2) = 108ab^2$
[This perfectly matches the third term of our rearranged polynomial.]
Step 5: Structural Mapping and Visualization
Since all four terms map perfectly to the $(x - y)^3$ identity, we can rewrite the entire polynomial in its unexpanded structural form.
$64a^3 - 144a^2b + 108ab^2 - 27b^3 = (4a)^3 - 3(4a)^2(3b) + 3(4a)(3b)^2 - (3b)^3$
Step 6: Final Factorization
By applying the identity $(x - y)^3$, we condense the expression into a single perfect cube:
$P(a,b) = (4a - 3b)^3$
To express this as a product of irreducible linear factors (which is the standard convention for complete factorization), we expand the exponent:
$P(a,b) = (4a - 3b)(4a - 3b)(4a - 3b)$
Final Solution: The completely factorised form of the polynomial $64a^3 - 27b^3 - 144a^2b + 108ab^2$ is $(4a - 3b)(4a - 3b)(4a - 3b)$.
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(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(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
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