Algebra

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.
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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