Find the best tutors and institutes for Class 10 Tuition
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
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Polynomials
Shankar Vihar, Delhi
18 yrs of
Exp
Student's Home
We are teaching 8th, 9th,10th,11th,12th for the last 12 years and providing 95%+ in our subjects. We are giving best results in DELHI CANTT and South Delhi region.
Top Tutor
Jahangirpura, Surat
15 yrs of
Exp
18 students
1954 hrs of classes completed
250
per hour
Online Classes
I love to teach students the concepts of Maths and Science. I believe in concept clearance rather than rote learning. Teaching is my passion. Many students have achieved 95 % and more in past years and and I believe that if you join and work hard sincerely, you can also achieved such scores.
Good teaching concept and clear explanation of concepts. Keeps students motivated and focused and ensures students understand the basics well. Very supportive and encouraging.
Top Tutor
Vesu, Surat
20 yrs of
Exp
18 students
1271 hrs of classes completed
500
per hour
Online Classes
I have experience teaching Mathematics to Grade 10 students with a focus on strengthening fundamental concepts and preparing them for board examinations. I teach important topics such as Real Numbers, Polynomials, Pair of Linear Equations in Two Variables, Quadratic Equations, Arithmetic Progressions, Trigonometry, Coordinate Geometry, Statistics, and Probability. My teaching approach involves explaining concepts clearly, solving problems step by step, and giving regular practice from NCERT and previous years’ board exam papers. I also conduct tests, revision classes, and doubt-clearing sessions to track students’ progress. My goal is to make mathematics simple and interesting while helping students build confidence and achieve good results in their Grade 10 board exams.
Very good teacher. Patient with the students and understands where they are confused quickly. Motivates the students to be better too.
Top Tutor
Sultanpur Viran, Sultanpur Viran
9 yrs of
Exp
9 students
1105 hrs of classes completed
1000
per hour
Online Classes
Student's Home
Tutor's home
I have a teaching experience of 9+ years. Currently teaching in St. Mary's Convent School. My way of teaching is simple and friendly as per the requirement. I lay emphasis on the basics because that only helps to make the concepts more easier to understand.
Top Tutor
Virupakshapura, Bangalore
19 yrs of
Exp
7 students
427 hrs of classes completed
500
per hour
Online Classes
Tutor's home
I've been teaching class 10 CBSE students for over 12 years now and have improved their ability to grasp concepts faster and implement the concepts through real examples. This method has been useful to ignite a liking for the subject and get better grades.
I'm really grateful to have Rajitha ma'am as my maths teacher. She teaches so well, she explains and clarifies doubts clearly that I'm now able to solve tough sums. The tricks you taught me have helped me in my exams. Thank you so much ma'am for helping me score better in my math exams.
Find more Tutor for Polynomials in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
