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Q8(iii):
Factorise each of the following:
(iii) $27 – 125a^3 – 135a + 225a^2$
Solution :
Initial Setup & Given Polynomial
We are tasked with factorising the following algebraic expression:
$P(a) = 27 - 125a^3 - 135a + 225a^2$
Step 1: Identifying the Relevant Algebraic Identity
By analyzing the polynomial, we observe the presence of perfect cubes ($27$ and $125a^3$) and alternating signs. This structural pattern strongly indicates the application of the standard cubic identity for the difference of two terms [Per the Binomial Theorem for exponent 3]:
$(x - y)^3 = x^3 - y^3 - 3x^2y + 3xy^2$
To utilize this identity, we must map the terms of our given polynomial to the terms of the expansion.
Step 2: Extracting the Base Terms (Cube Roots)
We isolate the perfect cube terms to determine our candidate values for $x$ and $y$.
- First term (Constant): The term $27$ is a perfect cube.
$27 = (3)^3 \implies x = 3$ - Second term (Cubic): The term $125a^3$ is a perfect cube.
$125a^3 = (5a)^3 \implies y = 5a$
Step 3: Verifying the Cross Terms
To rigorously confirm that the polynomial fits the identity $(x - y)^3$, we must evaluate the intermediate cross terms $-3x^2y$ and $+3xy^2$ using our candidate values $x = 3$ and $y = 5a$.
- Evaluating $-3x^2y$:
$-3(3)^2(5a) = -3(9)(5a) = -27(5a) = -135a$
[This perfectly matches the third term of the given polynomial]. - Evaluating $+3xy^2$:
$+3(3)(5a)^2 = 9(25a^2) = 225a^2$
[This perfectly matches the fourth term of the given polynomial].
Visual Mapping of the Algebraic Identity
The following diagram illustrates the exact one-to-one mapping between the given polynomial terms and the components of the cubic identity.
Step 4: Synthesizing the Factorised Form
Since all terms perfectly align with the expansion of $(x - y)^3$, we can rewrite the original polynomial in its expanded identity form:
$27 - 125a^3 - 135a + 225a^2 = (3)^3 - (5a)^3 - 3(3)^2(5a) + 3(3)(5a)^2$
Applying the identity $(x - y)^3$, where $x = 3$ and $y = 5a$, we condense the expression into a single perfect cube:
$= (3 - 5a)^3$
To express this as a product of its linear factors, we write the binomial three times.
Final Solution: The completely factorised form of the polynomial is $(3 - 5a)(3 - 5a)(3 - 5a)$.
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(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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