Polynomials Class 10 Maths NCERT Solutions NCERT Solutions for Class 10 Maths Chapter 2 Polynomials are part of NCERT Solutions for Class 10 Maths. Here we have given Maths NCERT Solutions Class 10 Chapter 2 Polynomials
Class 10 Maths NCERT Solutions Chapter 2 Polynomials
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.
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
.
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 ,
is a factor of .
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)
= 2()
=
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.
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: It 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
will be perfectly divisible by .
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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