This document discusses algebraic identities and factorizing polynomials. It begins by defining polynomials and their classification based on number of terms and degree. Some common algebraic identities involving addition, subtraction, and multiplication of polynomials are presented. These identities can be used to factorize polynomials. The document then describes an activity to verify the identity (a + b)3 = a3 + 3ab(a + b) + b3 using cubes and cuboids. A similar activity is presented for the identity (a - b)3 = a3 - 3ab(a - b) - b3.
Are you scared of that algebraic sums? Just view this presentation an you can learn about each and every algebraic identities. Just view this and Now take full marks in your tests..
Are you scared of that algebraic sums? Just view this presentation an you can learn about each and every algebraic identities. Just view this and Now take full marks in your tests..
Introduction to basic Mathematics , multiplication, division, addition and subtraction with concept of BODMAS .Basic operations fractions , decimals and percentage.
Introduction to basic Mathematics , multiplication, division, addition and subtraction with concept of BODMAS .Basic operations fractions , decimals and percentage.
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
this is the chapter-2 of class-10 Ncert Book..
polynomials..
You Can go through my profile and read ncert Chapter-1..
watch/read this book from --
|| Slides By MANAV ||
enjoy the formulas and use it with convidence and make your PT3 AND SPM more easier..togrther we achieve the better:)
good luck guys and girls...simple and short ans also sweet formulas..
APEX INSTITUTE was conceptualized in May 2008, keeping in view the dreams of young students by the vision & toil of Er. Shahid Iqbal. We had a very humble beginning as an institute for IIT-JEE / Medical, with a vision to provide an ideal launch pad for serious JEE students . We actually started to make a difference in the way students think and approach problems.
Synthetic Fiber Construction in lab .pptxPavel ( NSTU)
Synthetic fiber production is a fascinating and complex field that blends chemistry, engineering, and environmental science. By understanding these aspects, students can gain a comprehensive view of synthetic fiber production, its impact on society and the environment, and the potential for future innovations. Synthetic fibers play a crucial role in modern society, impacting various aspects of daily life, industry, and the environment. ynthetic fibers are integral to modern life, offering a range of benefits from cost-effectiveness and versatility to innovative applications and performance characteristics. While they pose environmental challenges, ongoing research and development aim to create more sustainable and eco-friendly alternatives. Understanding the importance of synthetic fibers helps in appreciating their role in the economy, industry, and daily life, while also emphasizing the need for sustainable practices and innovation.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
1. Activity Plus in Mathematics-9 25
Introduction
We have studied about a particular type of algebraic expression, called polynomial, and the terminology
related to it. Now, we shall study the factorisation of polynomials. In addition, we shall study some more
algebraic identities and their use in factorisation.
Polynomials in one variable: A polynomial p(x) in one variable x is an algebraic expression in x
of the form p(x) = an
xn
+ an – 1
xn – 1
+ ... + a2
x2
+ a1
x + a0
, where a0
, a1
, a2
, ...., an
are constants and
an
≠ 0. Here, a0
, a1
, a2
, ..., an
are respectively the coefficients of x0
, x, x2
, ..., xn
, and n (the highest power
is called the degree of the polynomial p(x). Each of an
xn
, an – 1
xn – 1
, ..., a0
with an
≠ 0, is called a term of
polynomial (px).
Polynomials are classified according to the number of their terms as well as according to their degree.
(i) A polynomial of one term is called a monomial.
Examples: 5x2
, 4x, –54x3
, etc.
(ii) A polynomial of two terms is called a binomial.
Examples: x + 1, x2
– x, y2
+ 1, etc.
(iii) A polynomial of three terms called a trinomial.
Examples: x2
+ x + 2, 2 2
+ −x x , etc.
(iv) A polynomial of degree 0 is called a constant polynomial.
Examples: 12, 74, –84, etc.
(v) A polynomial of degree 1 is called a linear polynomial.
Examples: 2x – 2, 2 1y + , etc.
(vi) A polynomial of degree 2 is called a quadratic polynomial.
Examples: 2x2
+ 5, 5x2
+ 3x, etc.
(vii) A polynomial of degree 3 is called a cubic polynomial.
Examples: 3x3
, 2x3
+ 1, 5x3
+ x2
, etc.
Algebraic identities: An algebraic identity is an algebraic equation that is true for all values of the variables
occurring in it. Algebraic identities are used to factorise algebraic expressions and also in computations.
Some algebraic identities are as follows:
(i) (a + b)2
= a2
+ 2ab + b2
(ii) (a – b)2
= a2
– 2ab + b2
(iii) a2
– b2
= (a + b)(a – b) (iv) (a + b + c)2
= a2
+ b2
+ c2
+ 2ab + 2bc + 2ca
(v) (a + b)3
= a3
+ b3
+ 3a2
b + 3ab2
(vi) (a – b)3
= a3
– b3
– 3ab(a – b)
(vii) a3
+ b3
= (a + b)(a2
– ab + b2
) (viii) a3
– b3
= (a – b)(a2
+ ab + b2
)
Multiplication of polynomials: To multiply two polynomials, multiply each term in one polynomial by
each term in the other polynomial and add those answers together. Simplify if needed.
Factorisation of polynomials: To factorise quadratic polynomials of the type ax2
+ bx + c, where a ≠ 0
and a, b, c are constants, we have to write b as the sum of two numbers whose product is ac.
Algebra2
2. Activity Plus in Mathematics-926
Activity 2.1 Algebraic Identity : (a + b)3
= a3
+ 3ab(a + b) + b3
To compute (a + b)3
, we extend the identity (a + b)2
= a2
+ 2ab + b2
as follows:
(a + b)3
= (a + b)(a + b)2
= (a + b)(a2
+ 2ab + b2
)
= a(a2
+ 2ab + b2
) + b(a2
+ 2ab + b 2
) = a3
+ 2a2
b + ab2
+ a2
b + 2ab2
+ b3
= a3
+ 3a2
b + 3ab2
+ b3
= a3
+ 3ab(a + b) + b3
Objective
To verify the algebraic identity:
(a + b)3
= a 3
+ 3ab(a + b) + b 3
.
Pre-requisite knowledge
(i ) Concept of a cube and a cuboid
(ii ) Volume of a cube = Side × Side × Side
(iii ) Volume of a cuboid = Length × Breadth × Height
Procedure
(i ) Make a cube of side ‘a’ units as shown in the figure. Its volume is a3
. Wrap a red glazed paper on it.
Fig. 1
(ii ) Make another cube of side ‘b’ units as shown in the figure. Its volume is b3
. Wrap a black glazed
paper on it.
Fig. 2
(iii ) Make three cuboids each of dimensions (a) × (b) × (a + b) units. Wrap them with green glazed papers
as shown in the figure. Volume of each cuboid is a · b(a + b) cubic units.
Fig. 3
Materials Required
Cardboard
Glazed paper of various colours
Sheets of white paper
A pair of scissors
Scale, Cutter, Sketch pens
Geometry box, Fevicol
3. Activity Plus in Mathematics-9 27
(iv ) Arrange the above two cubes and three cuboids in such a way that they altogether make a cube as
shown in the figure.
Fig. 4
(v ) A cubed binomial (sum) is equal to the cube of the first, plus three times the square of the first by
the second, plus three times the first by the square of the second, plus the cube of the second.
Observations
(i ) We have joined the above five blocks (two cubes and three cuboids) to form a cube of side (a + b) units.
(ii ) The volume of this cube is (a + b)3
cubic units.
(iii ) The volume of the two cubes are a 3
and b3
cubic units.
(iv ) The volume of each of the cuboid is ab(a + b) cubic units.
i.e., the volume of all the three cuboids is 3ab(a + b) cubic units.
Conclusion
⇒ (a + b)3
= a3
+ b3
+ 3[ab(a + b)]
or (a + b)3
= a3
+ 3ab(a + b) + b3
Learning Outcomes
(i ) The students will obtain the skill of adding the volumes of cubes and cuboids.
(ii ) Showing the volume of a cube as the sum of cubes and cuboids helps the students to get a geometric
feeling of volume.
Remark
The identity (a + b)3
= a3
+ b3
+ 3ab(a + b) is useful for calculating the cube of a number which can be
expressed as the sum of two convenient numbers.
The blocks used in this activity can be cut-out from soft-wood or can be made using cardboard or
thermocol sheet.
Note:
4. Activity Plus in Mathematics-928
Suggested Activity
1. To verify that (y + z)3
= y3
+ z3
+ 3y2
z + 3yz2
by taking y = 14 and z = 4.
Viva Voce
Q1. In the activity of (a + b)3
, what do you mean by 3a2
b, 3ab2
?
Ans. 3a2
b represents sets of cuboids with volumes a × a × b.
And, 3ab2
represents three sets of cuboids with volumes a × b × b.
Q2. If one side of a cube is (a + 2b), then what is the volume of the cube?
Ans. Volume of cube = (a + 2b)3
.
Q3. What is the degree of (5x + 7y)3
?
Ans. 3
Q4. What is the coefficient of z2
in (3y +4z)3
?
Ans. 144y.
Q5. Is (a + b)3
binomial?
Ans. No.
Q6. What is the value of (x + 1)3
?
Ans. x3
+ 3x2
+ 3x + 1.
Q7. What is an equation?
Ans. A statement of equality is called an equation.
Q8. What is the difference between an equation and an identity?
Ans. An equation is not true for all values of the variable whereas an identity is true for every value of
the variable in it.
Q9. The L.H.S. of an identity is (a + b)3
. What is its R.H.S.?
Ans. a 3
+ 3a 2
b + 3ab 2
+ b 3
.
Q10. What is the R.H.S. of the identity whose L.H.S. is (a + 2)3
?
Ans. a 3
+ 6a2
+ 12a + 8.
qqq
5. Activity Plus in Mathematics-9 29
Activity 2.2 Algebraic Identity : (a – b)3
= a3
– 3ab(a – b) – b3
To compute (a – b)3
, we extend the identity (a – b)2
= a2
– 2ab + b2
as follows:
(a – b)3
= (a – b)(a – b)2
= (a – b)(a2
– 2ab + b2
)
= a(a2
– 2ab + b2
) – b(a2
– 2ab + b2
) = a3
– 2a2
b + ab2
– a2
b + 2ab2
– b3
= a3
– 3a2
b + 3ab2
– b3
= a3
– 3ab(a – b) – b3
Objective
To verify the algebraic identity:
(a – b)3
= a3
– 3ab(a – b) – b3
.
Pre-requisite Knowledge
(i ) Volume of a cube = Side × Side × Side
(ii ) Volume of a cuboid = Length × Breadth × Height
Procedure
(i ) Make a cube of ‘b’ units as shown in figure-1. Its volume is b3
cubic units. Wrap a red glazed paper
on it.
(ii ) Make another cube of side (a – b) units as shown in figure 2 and wrap a black glazed paper on it.
Its volume is (a – b)3
cubic units.
Fig. 1 Fig. 2
(iii ) Make three cuboids each of dimensions ‘a’, ‘b’ and (a – b) units as shown in figure 3. Volume of
each cuboid is a × b × (a – b) cubic units. Wrap them with green glazed paper.
Fig. 3
Materials Required
Cardboard
Glazed paper of red, green and
black colours
Sketch pen, Geometry box
Fevicol, Scale, Cutter, White papers
A pair of scissors, Cello tape
6. Activity Plus in Mathematics-930
(iv ) Arrange the above five blocks (two cubes and three cuboids) in such a manner that they form a big
cube as shown below:
Fig. 4
Observations
(i ) Each side of the resultant cube is ‘a’ units. [ (a – b) + b = a]
(ii ) Volume of this cube = a3
cubic units.
(iii ) This cube is formed by joining the five blocks (2 cubes and 3 cuboids).
(iv ) The volume of the two cubes are (a – b)3
cubic units and b3
cubic units.
(v ) The volume of the three cuboids taken together is 3ab(a – b) cubic units.
(vi ) From the cube of side ‘a’ units, if we remove a cube of volume b3
cubic units (wrapped by a red
glazed paper) and the three cuboids each of volume ab(a – b) cubic units (wrapped by a green glazed
paper), then we are left with a cube of side (a – b) units.
Conclusion
⇒ (a – b)3
+ b3
+ 3[ab(a – b)] = a3
or (a – b)3
= a3
– 3ab(a – b) – b3
.
Learning Outcomes
(i ) The students will obtain the skill of adding the volume of cubes and cuboids.
(ii ) The students will obtain the skill of making a big cube using cuboids and cubes.
(iii ) Showing the volume of a cube as the sum of cubes and cuboids helps them to get a geometric feeling
of volume.
Remark
We call the right hand side expression, i.e., a3
– 3ab(a – b) – b3
the expanded form of the left hand side
expression, i.e., (a – b)3
.
7. Activity Plus in Mathematics-9 31
Suggested Activity
1. To verify that (y – z)3
= y3
– 3y2
z + 3yz2
– z3
by taking y = 12 and z = 2.
Viva Voce
Q1. What is the maximum number of zeros that a cubic polynomial can have?
Ans. Three.
Q2. Expand (x – 3y)3
.
Ans. x3
– 27y3
– 9x2
y + 27xy2
.
Q3. For evaluating (999)3
, which formula we should use?
Ans. We should use
(a – b)3
= a3
– b3
– 3ab(a – b) by taking a = 1000 and b = 1.
Q4. How would you expand p3
– q3
in terms of (p – q)3
?
Ans. We know that
(p – q)3
= p3
– q3
– 3pq(p – q)
= p3
– q3
– 3p2
q + 3pq2
⇒ p3
– q3
= (p – q)3
+ 3p2
q – 3pq2
.
Q5. If one side of a cube is given by (a – b), then what is the volume of the cube?
Ans. Volume of cube (a – b)3
.
Q6. Write the coefficient of x3
in (5x – 3y)3
.
Ans. 125.
Q7. The L.H.S. of an identity is (a – b)3
. What is its R.H.S.?
Ans. a 3
– 3a 2
b + 3ab 2
– b 3
.
Q8. What is the RHS of the identity whose L.H.S. is (a – 1)3
?
Ans. a 3
– 3a 2
+ 3a – 1.
Q9. What is the short form of 8x3
+ 27y3
+ 36x2
y + 54xy2
?
Ans. 8x3
+ 27y3
+ 36x2
y + 54xy2
= (2x)3
+ (3y)3
+ 3(4x
2
)(3y) + 3(2x)(9y
2
)
= (2x)3
+ (3y)3
+ 3(2x)2
(3y) + 3(2x)(3y)2
= (2x + 3y)3
qqq