Chapter 2 – Polynomials
Exercise 2.1
Question 1:
The graphs of y = p(x) are given in following figure, for some polynomials p(x). Find the number of zeroes of p(x), in each case.
(i)

(ii)

(iii)

(iv)

(v)

(v)

Answer:
(i) The number of zeroes is 0 as the graph does not cut the x-axis at any point.
(ii) The number of zeroes is 1 as the graph intersects the x-axis at only 1 point.
(iii) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.
(iv) The number of zeroes is 2 as the graph intersects the x-axis at 2 points.
(v) The number of zeroes is 4 as the graph intersects the x-axis at 4 points.
(vi) The number of zeroes is 3 as the graph intersects the x-axis at 3 points.
Exercise 2.2
Question 1:
Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.






Answer:

The value of
is zero when x − 4 = 0 or x + 2 = 0, i.e., when x = 4 or x = −2

Therefore, the zeroes of
are 4 and −2.

Sum of zeroes = 

Product of zeroes 


The value of 4s2 − 4s + 1 is zero when 2s − 1 = 0, i.e.,

Therefore, the zeroes of 4s2 − 4s + 1 are
and
.


Sum of zeroes = 

Product of zeroes 


The value of 6x2 − 3 − 7x is zero when 3x + 1 = 0 or 2x − 3 = 0, i.e.,
or


Therefore, the zeroes of 6x2 − 3 − 7x are
.

Sum of zeroes = 

Product of zeroes = 


The value of 4u2 + 8u is zero when 4u = 0 or u + 2 = 0, i.e., u = 0 or u = −2
Therefore, the zeroes of 4u2 + 8u are 0 and −2.
Sum of zeroes = 

Product of zeroes = 


The value of t2 − 15 is zero when
or
, i.e., when 



Therefore, the zeroes of t2 − 15 are
and
.


Sum of zeroes =

Product of zeroes = 


The value of 3x2 − x − 4 is zero when 3x − 4 = 0 or x + 1 = 0, i.e., when
or x = −1

Therefore, the zeroes of 3x2 − x − 4 are
and −1.

Sum of zeroes = 

Product of zeroes 

Question 2:
Find a quadratic polynomial each with the given numbers as the sum and product of its zeroes respectively.






Answer:

Let the polynomial be
, and its zeroes be
and
.




Therefore, the quadratic polynomial is 4x2 − x − 4.

Let the polynomial be
, and its zeroes be
and
.




Therefore, the quadratic polynomial is 3x2 −
x + 1.


Let the polynomial be
, and its zeroes be
and
.




Therefore, the quadratic polynomial is
.


Let the polynomial be
, and its zeroes be
and
.




Therefore, the quadratic polynomial is
.


Let the polynomial be
, and its zeroes be
and
.




Therefore, the quadratic polynomial is
.


Let the polynomial be
.


Therefore, the quadratic polynomial is
.

Exercise 2.3
Question 1:
Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following:
(i) 

(ii) 

(iii) 

Answer:


Quotient = x − 3
Remainder = 7x − 9


Quotient = x2 + x − 3
Remainder = 8


Quotient = −x2 − 2
Remainder = −5x +10
Question 2:
Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

Answer:




Since the remainder is 0,
Hence,
is a factor of
.




Since the remainder is 0,
Hence,
is a factor of
.




Since the remainder
,

Hence,
is not a factor of
.


Question 3:
Obtain all other zeroes of
, if two of its zeroes are
.


Answer:

Since the two zeroes are
,



Therefore, we divide the given polynomial by
.


We factorize 


Therefore, its zero is given by x + 1 = 0
x = −1
As it has the term
, therefore, there will be 2 zeroes at x = −1.

Hence, the zeroes of the given polynomial are
, −1 and −1.

Question 4:
On dividing
by a polynomial g(x), the quotient and remainder were x − 2 and − 2x + 4, respectively. Find g(x).

Answer:

g(x) = ? (Divisor)
Quotient = (x − 2)
Remainder = (− 2x + 4)
Dividend = Divisor × Quotient + Remainder

g(x) is the quotient when we divide
by




Question 5:
Give examples of polynomial p(x), g(x), q(x) and r(x), which satisfy the division algorithm and
(i) deg p(x) = deg q(x)
(ii) deg q(x) = deg r(x)
(iii) deg r(x) = 0
Answer:
According to the division algorithm, if p(x) and g(x) are two polynomials with
g(x) ≠ 0, then we can find polynomials q(x) and r(x) such that
p(x) = g(x) × q(x) + r(x),
where r(x) = 0 or degree of r(x) < degree of g(x)
Degree of a polynomial is the highest power of the variable in the polynomial.
(i) deg p(x) = deg q(x)
Degree of quotient will be equal to degree of dividend when divisor is constant ( i.e., when any polynomial is divided by a constant).
Let us assume the division of
by 2.

Here, p(x) = 

g(x) = 2
q(x) =
and r(x) = 0

Degree of p(x) and q(x) is the same i.e., 2.
Checking for division algorithm,
p(x) = g(x) × q(x) + r(x)


= 

Thus, the division algorithm is satisfied.
(ii) deg q(x) = deg r(x)
Let us assume the division of x3 + x by x2,
Here, p(x) = x3 + x
g(x) = x2
q(x) = x and r(x) = x
Clearly, the degree of q(x) and r(x) is the same i.e., 1.
Checking for division algorithm,
p(x) = g(x) × q(x) + r(x)
x3 + x = (x2 ) × x + x
x3 + x = x3 + x
Thus, the division algorithm is satisfied.
(iii)deg r(x) = 0
Degree of remainder will be 0 when remainder comes to a constant.
Let us assume the division of x3 + 1by x2.
Here, p(x) = x3 + 1
g(x) = x2
q(x) = x and r(x) = 1
Clearly, the degree of r(x) is 0.
Checking for division algorithm,
p(x) = g(x) × q(x) + r(x)
x3 + 1 = (x2 ) × x + 1
x3 + 1 = x3 + 1
Thus, the division algorithm is satisfied.
Exercise 2.4
Question 1:
Verify that the numbers given alongside of the cubic polynomials below are their zeroes. Also verify the relationship between the zeroes and the coefficients in each case:

Answer:
(i) 



Therefore,
, 1, and −2 are the zeroes of the given polynomial.

Comparing the given polynomial with
, we obtain a = 2, b = 1, c = −5, d = 2


Therefore, the relationship between the zeroes and the coefficients is verified.
(ii) 



Therefore, 2, 1, 1 are the zeroes of the given polynomial.
Comparing the given polynomial with
, we obtain a = 1, b = −4, c = 5, d = −2.

Verification of the relationship between zeroes and coefficient of the given polynomial

Multiplication of zeroes taking two at a time = (2)(1) + (1)(1) + (2)(1) =2 + 1 + 2 = 5 

Multiplication of zeroes = 2 × 1 × 1 = 2 

Hence, the relationship between the zeroes and the coefficients is verified.
Question 2:
Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, − 7, − 14 respectively.
Answer:
Let the polynomial be
and the zeroes be
.


It is given that

If a = 1, then b = −2, c = −7, d = 14
Hence, the polynomial is
.

Question 3:
If the zeroes of polynomial
are
, find a and b.


Answer:

Zeroes are a − b, a + a + b
Comparing the given polynomial with
, we obtain

p = 1, q = −3, r = 1, t = 1

The zeroes are
.


Hence, a = 1 and b =
or
.


Question 4:
If two zeroes of the polynomial
are
, find other zeroes.


Answer:
Given that 2 +
and 2
are zeroes of the given polynomial.



Therefore,
= x2 + 4 − 4x − 3

= x2 − 4x + 1 is a factor of the given polynomial
For finding the remaining zeroes of the given polynomial, we will find the quotient by dividing
by x2 − 4x + 1.


Clearly,
= 


It can be observed that
is also a factor of the given polynomial.

And
= 


Therefore, the value of the polynomial is also zero when
or 


Or x = 7 or −5
Hence, 7 and −5 are also zeroes of this polynomial.
Question 5:
If the polynomial
is divided by another polynomial
, the remainder comes out to be x + a, find k and a.


Answer:
By division algorithm,
Dividend = Divisor × Quotient + Remainder
Dividend − Remainder = Divisor × Quotient


Let us divide
by 



It can be observed that
will be 0.

Therefore,
= 0 and
= 0


For
= 0,

2 k =10
And thus, k = 5
For
= 0

10 − a − 8 × 5 + 25 = 0
10 − a − 40 + 25 = 0
− 5 − a = 0
Therefore, a = −5
Hence, k = 5 and a = −5
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