NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Exercise 2.1

1. The graphs of y = p(x) are given in Fig. 2.10 below, for some polynomials p(x). Find the number of zeroes of p(x), in each case.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Solution

We can find the number of zeroes p(x) has by counting the points of intersection of the graph on X-axis. By applying this knowledge, we can conclude that the graphs have following numbers of zeroes:

  1. No zeroes
  2. One zero
  3. Three zeroes
  4. Two zeroes
  5. Four zeroes
  6. Three zeroes

2. Find the zeroes of the following quadratic polynomials and verify the relationship between the zeroes and the coefficients.

  1. x2 - 2x - 8
  2. 4s2 - 4s + 1
  3. 6x2 - 3 - 7x
  4. 4u2 + 8u
  5. t2 - 15
  6. 3x2 - x - 4

Solution

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Hence, the given polynomial has two zeroes.

Verification

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

II.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Hence, the given polynomial has two zeroes.

Verification

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

III.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Hence, the given polynomial has two zeroes.

Verification

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

IV.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Hence, the given polynomial has two zeroes.

Verification

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

V.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Hence, the given polynomial has two zeroes.

Verification

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

VI.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Hence, the given polynomial has two zeroes.

Verification

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

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

  1. 1/4 , -1
  2. √2, 1/3
  3. 0, √5
  4. 1, 1
  5. -1/4, 1/4
  6. 4, 1

Solution

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Exercise 2.3

1. Divide the polynomial p(x) by the polynomial g(x) and find the quotient and remainder in each of the following :

  1. p(x) = x3 - 3x2 + 5x - 3, g(x) = x2 - 2
  2. p(x) = x4 - 3x2 + 4x + 5, g(x) = x2 + 1 - x
  3. p(x) = x4 - 5x + 6, g(x) = 2 - x2

Solution

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

2. Check whether the first polynomial is a factor of the second polynomial by dividing the second polynomial by the first polynomial:

  1. t2 - 3, 2t4 + 3t3 - 2t2 - 9t - 12
  2. x2 + 3x + 1, 3x4 + 5x3 - 7x2 + 2x + 2
  3. x3 - 3x + 1, x5 - 4x3 + x2 + 3x + 1

Solution

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Since, the remainder is zero.

Therefore, first polynomial is a factor of the second polynomial.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Since, the remainder is zero.

Therefore, first polynomial is a factor of the second polynomial.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Since, the remainder is not zero.

Therefore, first polynomial is not a factor of the second polynomial.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Solution

Putting the two zeroes in (x - a)(x - b)

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Now, 1/3 and 3x2 - 5, both are factors of the given polynomial. If we divide 3x2 - 5 by the given polynomial, we get:

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Therefore,

3x4 + 6x3 - 2x2 - 10x - 5 = (3x2 - 5) (x2 + 2x + 1)

= (3x2 - 5) [(x2 + x + x + 1)]

= (3x2 - 5) [x(x + 1) + 1(x + 1)]

= (3x2 - 5) (x + 1) (x + 1)

When (x + 1) = 0 � x = -1 and when (x + 1) = 0 � x = -1.

Hence, the remaining two zeroes of the given polynomial are -1 and -1.

4. On dividing x3 - 3x2 + x + 2 by a polynomial g(x), the quotient and remainder were x - 2 and -2x + 4, respectively. Find g(x).

Solution

We know that,

p(x) = g(x) × Quotient + Remainder

Therefore,

x3 - 3x2 + x + 2 = g(x) × (x - 2) + (-2x + 4)

x3 - 3x2 + x + 2 + 2x - 4= g(x) × (x - 2)

x3 - 3x2 + 3x - 2 = g(x) × (x - 2)

On dividing x - 2 with x3 - 3x2 + 3x - 2, we get

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

5. Give examples of polynomials p(x), g(x), q(x) and r(x), which satisfy the division algorithm and:

  1. deg p(x) = deg q(x)
  2. deg q(x) = deg r(x)
  3. deg r(x) = 0

Solution

I. p(x) = 8x2 + 4x + 2

q(x) = 4x2 + 2x + 1

g(x) = 2

r(x) = 0

II. p(x) = x3 - 3x2 + 5x - 3

q(x) = x - 3

g(x) = x2 - 2

r(x) = 7x - 9

III. p(x) = x3 - x2 + 2x + 3

q(x) = x - 1

g(x) = x2 + 2

r(x) = 5

Exercise 2.4 (Optional)

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:

  1. 2x3 + x2 - 5x + 2; 1/2 , 1, - 2
  2. x3 - 4x2 + 5x - 2; 2, 1, 1

Solution

I. Comparing the given polynomial with p(x) = ax3 + bx2 + cx +d, we get

a = 2, b = 1, c = -5, d = 2

Now we will check if the given numbers are zeroes of the given polynomial by substituting them for x.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Hence, the given numbers are the zeroes of the given polynomial.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

II. Comparing the given polynomial with p(x) = ax3 + bx2 + cx +d, we get

a = 1 , b = -4, c = 5, d = -2

Now we will check if the given numbers are zeroes of the given polynomial by substituting them for x.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Hence, the given numbers are the zeroes of the given polynomial.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

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.

Solution

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

3. If the zeroes of the polynomial x3 - 3x2 + x + 1 are a - b, a, a + b, find a and b.

Solution

Comparing the given polynomial with Ax3 + Bx2 + Cx + D, we get

A = 1, B = -3, C = 1, and D = 1

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

4. If two zeroes of the polynomial x4 - 6x3 - 26x2 + 138x - 35 are 2 � √3, find other zeroes.

Solution

Substituting the two zeroes in (x - a) (x - b)

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Now, x2 - 4x + 1 is a factor of the given polynomial. If we divide x2 - 4x + 1 by the given polynomial, we get:

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Therefore,

x4 - 6x3 - 26x2 + 138x - 35 = (x2 - 4x + 1) (x2 -� 2x - 35)

= (x2 - 4x + 1) (x2 - 7x + 5x - 35)

= (x2 - 4x + 1) {x(x - 7) + 5(x - 7)}

= (x2 - 4x + 1) (x - 7) (x + 5)

Hence, the two remaining zeroes will be 7 and -5.

5. If the polynomial x4 - 6x3 + 16x2 - 25x + 10 is divided by another polynomial x2 - 2x + k, the remainder comes out to be x + a, find k and a.

Solution

First, we need to divide x4 - 6x3 + 16x2 - 25x + 10 by x2 - 2x + k and find out the remainder.

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Upon comparing the coefficients, we get two equations

First equation -

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Second equation -

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Substitute k = 5

NCERT Solutions Class 10 Maths Chapter-2 : Polynomials

Hence, k = 5 and a = -5.






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