Find the best tutors and institutes for Class 10 Tuition
Q9(ii):
Verify : (ii) $x^3 – y^3 = (x – y) (x^2 + xy + y^2)$
Solution :
Initial Setup & Objective
We are tasked with verifying the fundamental algebraic identity for the difference of two cubes. The identity is given as:
$x^3 - y^3 = (x - y)(x^2 + xy + y^2)$
To verify this rigorously, we will expand the Right-Hand Side (RHS) and demonstrate that it simplifies exactly to the Left-Hand Side (LHS). We will also provide a secondary derivation using the binomial cube expansion to establish deep theoretical consistency.
Method 1: Direct Algebraic Expansion
Step 1: Applying the Distributive Property
We begin with the expression on the Right-Hand Side (RHS):
$RHS = (x - y)(x^2 + xy + y^2)$
[Per the Distributive Law of Multiplication over Addition, $A(B + C + D) = AB + AC + AD$]. We must distribute both terms of the binomial $(x)$ and $(-y)$ across the entire trinomial:
$RHS = x(x^2 + xy + y^2) - y(x^2 + xy + y^2)$
Step 2: Term-by-Term Multiplication
Next, we distribute $x$ into the first trinomial and $-y$ into the second trinomial. [By the laws of exponents, $x^a \cdot x^b = x^{a+b}$]:
- Multiplying by $x$:
$x \cdot x^2 = x^3$
$x \cdot xy = x^2y$
$x \cdot y^2 = xy^2$ - Multiplying by $-y$:
$-y \cdot x^2 = -x^2y$ [Note: Commutativity allows us to write $-yx^2$ as $-x^2y$]
$-y \cdot xy = -xy^2$
$-y \cdot y^2 = -y^3$
Substituting these expanded terms back into our equation yields:
$RHS = (x^3 + x^2y + xy^2) + (-x^2y - xy^2 - y^3)$
Step 3: Aggregation and Simplification
We now drop the parentheses and group the like terms together to identify additive inverses (terms that sum to zero):
$RHS = x^3 + x^2y - x^2y + xy^2 - xy^2 - y^3$
Rearranging the terms for absolute clarity:
$RHS = x^3 + (x^2y - x^2y) + (xy^2 - xy^2) - y^3$
The terms $x^2y$ and $-x^2y$ cancel each other out. Similarly, $xy^2$ and $-xy^2$ cancel each other out:
$RHS = x^3 + 0 + 0 - y^3$
$RHS = x^3 - y^3$
Since the simplified RHS perfectly matches the LHS, the identity is verified.
Visualizing the Expansion (Area Model Matrix)
The following matrix visually maps the distribution of $(x - y)$ across $(x^2 + xy + y^2)$. The color-coded cells demonstrate how the intermediate terms cancel each other out, leaving only the cubic terms.
Method 2: Derivation from the Binomial Cube Identity
To provide exhaustive proof, we can also derive this factorization directly from the standard identity for the cube of a binomial:
$ (x - y)^3 = x^3 - y^3 - 3xy(x - y) $
Step 1: Isolate the Difference of Cubes
We want to solve for $x^3 - y^3$. By adding $3xy(x - y)$ to both sides of the equation, we get:
$ x^3 - y^3 = (x - y)^3 + 3xy(x - y) $
Step 2: Factor out the Common Binomial
Observe that both terms on the right side share a common factor of $(x - y)$. We extract this factor:
$ x^3 - y^3 = (x - y) \left[ (x - y)^2 + 3xy \right] $
Step 3: Expand and Simplify the Inner Bracket
We expand the squared binomial $(x - y)^2 = x^2 - 2xy + y^2$ inside the brackets:
$ x^3 - y^3 = (x - y) \left[ (x^2 - 2xy + y^2) + 3xy \right] $
Combine the like terms ($-2xy$ and $+3xy$):
$ -2xy + 3xy = +xy $
Substituting this back into the bracket yields the final verified identity:
$ x^3 - y^3 = (x - y)(x^2 + xy + y^2) $
Final Solution: Through both direct algebraic expansion and derivation from the binomial cube identity, it is rigorously proven that $x^3 - y^3 = (x - y)(x^2 + xy + y^2)$. The intermediate terms $x^2y$ and $xy^2$ act as additive inverses and cancel out perfectly.
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(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) $x^3 + y^3 = (x + y) (x^2 – xy + y^2)$
- 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
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Polynomials
Top Tutor
Virugambakkam, Chennai
20 yrs of
Exp
36 students
1713 hrs of classes completed
500
per hour
Online Classes
I am R.Vinotha ., M.Sc.,M.Phil.,B.Ed in mathematics.I am the trainer/tutor of this course .I was allotted 12 regular classes for a month.I am using through google meet app for online class and also i am taking mathematics tuition at my home.I will create seperate WhatApp group for my tuition students and communicates to them .I will send the class notes to them after completing the class .My class planner is us following: Regular classes : Any 3 days for a week Class duration : 1 hour Home works: Daily write two problems and send it to be in WhatsApp group Test: Daily test : Formulae test: Everyday i am asking 10 formulae through voice message in WhatsApp group after getting my message ,the students will answer the questions and send it tome via tuition group Problems test:Daily write two problems Regular test : Every saturday and sunday Monthly test : Every month fourth saturday doubts clarification time : Every day 7 pm -8 pm I will correct the test papers immedietly and send it to their personal windows and also resolve the students mistakes. Chapter's quiz : After completing every chapters ,i will conduct mathematical quiz for that chapters Revisions: I will revise previous years questio papers After this type of my coachings many of my students got 100 marks in theire board examination .I put lots of efforts to my students and they getting good marks in the board examination .My teaching type is friendly manner.I gave extra solved questions and answers to my students .
Taking class as well as gives coaching in different approaches that can be dealt with the topic thro variety in problems and helps to reach solution with clarity of the nuances to be careful with.
Top Tutor
Nerul, Mumbai
10 yrs of
Exp
5 students
640 hrs of classes completed
450
per hour
Online Classes
Student's Home
Tutor's home
I have specifically guided Class 10 students for board exams, with a strong focus on NCERT syllabus, and topics like Real Numbers, Polynomials, Quadratic Equations, and Statistics. I provide regular tests, personalized feedback, and exam-oriented strategies to improve accuracy and speed. Many of my students have shown remarkable improvement and scored above 90% in their board exams.
It was an amazing experience learning and understanding maths with Saurav sir. He really helped me overcome my fear of maths and helped me score good marks in a subject that previously scared me. It really shows how dedicated he is as a teacher. Really good experience!
Top Tutor
Kushaiguda, Hyderabad
13 yrs of
Exp
1 student
126 hrs of classes completed
300
per hour
Online Classes
Student's Home
Teaching is my passion I am into teaching profession since 16 years. I like challenges with young minds.. I will make students to confident by explaining concepts, taking special care. Cent percent assurance to child growth, committed to work towards target. I used to counsel the students to succeed in their exams.
Top Tutor
Sector 43, Gurgaon
12 yrs of
Exp
1000
per hour
Online Classes
Student's Home
Experienced Mathematics and Science Educator | 10+ Years in Teaching and Coaching I have over a decade of experience in teaching and coaching students in Mathematics and Science. I have worked as a PGT Mathematics teacher at GBL Convent School, and have also taught Classes 9 and 10 (Maths & Science) at R.S. Education Hub. My teaching approach focuses on building strong conceptual clarity, personalized attention, and result-oriented strategies to help students excel academically. I have successfully guided numerous students through school exams, board exams, and foundational competitive preparation.
Top Tutor
Bajaj Nagar, Nagpur
3 yrs of
Exp
1 student
Online Classes
Student's Home
Coaching is about improvising a student from where he/she is at. Not about converting an already bright student into an achiever. It should be obvious. The whole idea of ONE ON ONE coaching is to improvise and to elimate the problems in a detailed manner, exactly the way a student desires. To connect and respond to a student cannot be everybodys cup of tea. This is where a capable one comes in to fill the gaps where modern day educationalists fail to fill.
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
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
