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Q8(i):
Factorise each of the following:
(i) $8a^3 + b^3 + 12a^2b + 6ab^2$
Solution :
Initial Algebraic Setup
We are tasked with factorising the following algebraic expression:
$8a^3 + b^3 + 12a^2b + 6ab^2$
Step 1: Identifying the Relevant Algebraic Identity
By analyzing the structure of the polynomial, we observe that it contains four terms, all of which are positive. Furthermore, the first and second terms ($8a^3$ and $b^3$) are perfect cubes. This structural signature strongly indicates the application of the standard binomial expansion identity for the cube of a sum [Per the Binomial Theorem for exponent 3].
The standard identity is defined as:
$(x + y)^3 = x^3 + y^3 + 3x^2y + 3xy^2$
Our objective is to map the given polynomial to the right-hand side of this identity.
Step 2: Decomposing the Terms to Establish Base Variables
We must rewrite the perfect cube terms to identify the base variables $x$ and $y$.
- First Term (Cube): The term $8a^3$ can be rewritten as the cube of a single monomial. Since $2^3 = 8$, we have:
$8a^3 = (2a)^3$
[This establishes our $x = 2a$]. - Second Term (Cube): The term $b^3$ is already a perfect cube:
$b^3 = (b)^3$
[This establishes our $y = b$].
Step 3: Verifying the Cross-Terms
To rigorously prove that the identity applies, we must verify that the remaining terms in the polynomial ($12a^2b$ and $6ab^2$) perfectly match the $3x^2y$ and $3xy^2$ components of the identity using our established $x = 2a$ and $y = b$.
- Checking $3x^2y$:
Substitute $x = 2a$ and $y = b$:
$3(2a)^2(b) = 3(4a^2)(b) = 12a^2b$
[This perfectly matches the third term of our given polynomial]. - Checking $3xy^2$:
Substitute $x = 2a$ and $y = b$:
$3(2a)(b)^2 = 3(2a)(b^2) = 6ab^2$
[This perfectly matches the fourth term of our given polynomial].
Step 4: Visualizing the Algebraic Mapping
The following diagram illustrates the exact one-to-one mapping between the standard identity and our decomposed polynomial.
Step 5: Applying the Identity to Factorise
Since all terms perfectly satisfy the expansion of $(x + y)^3$, we can condense the expanded polynomial into its factorised binomial cube form by substituting $x = 2a$ and $y = b$:
$(2a)^3 + (b)^3 + 3(2a)^2(b) + 3(2a)(b)^2 = (2a + b)^3$
Step 6: Expanding into Linear Factors
To express the polynomial fully factorised into irreducible linear polynomials, we expand the exponent. [By the fundamental definition of exponents, $z^3 = z \cdot z \cdot z$].
$(2a + b)^3 = (2a + b)(2a + b)(2a + b)$
Final Solution: The completely factorised form of the polynomial $8a^3 + b^3 + 12a^2b + 6ab^2$ is $(2a + b)(2a + b)(2a + b)$.
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) $(3 – 2x) (3 + 2x)$
- 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(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) $x^3 – y^3 = (x – y) (x^2 + xy + y^2)$
- 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 : $12ky^2 + 8ky – 20k$
CBSE Solutions for Class 9 Mathematics Polynomials
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