RS Agarwal Solution | Class 10th | Chapter-2 | Polynomials | Edugrown

Exercise Ex. 2A

Question 1

Find the zeros of the quadratic polynomial (x² + 3x – 10) and verify the relation between its zeros and coefficients.Solution 1

Question 2

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

x² – 2x – 8Solution 2

Question 3

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

x² + 7x + 12Solution 3

Question 5

Find the zeros of the quadratic polynomial (4x² – 4x – 3) and verify the relation between its zeros and coefficients.Solution 5

We have

Question 6

Find the zeros of the quadratic polynomial (5x² – 4 – 8x) and verify the relationship between its zeros and coefficients of the given polynomial.Solution 6

Question 7

Find the zeros of the quadratic polynomial (2x² – 11x + 15) and verify the relation between its zeros and coefficients.Solution 7

Question 8

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

Solution 8

Question 9

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

4x² – 4x + 1Solution 9

Question 10

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

3x² – x – 4Solution 10

Question 11

Find the zeros of the following quadratic polynomials and verify the relationship between the zeros and the coefficients:

5y² + 10ySolution 11

Question 12

Find the zeros of the quadratic polynomial (8x² – 4) and verify the relation between its zeros and coefficients.Solution 12

Let

Question 14

Find the quadratic polynomial, sum of whose zeros is 8 and their product is 12. Hence, find the zeros of the polynomial.Solution 14

Question 15

Find the quadratic polynomial, sum of whose zeros is 0 and their product is -1. Hence, find the zeros of the polynomial.Solution 15

Question 16

Find the quadratic polynomial, sum of whose zeros is  and their product is 1. Hence, find the zeros of the polynomial.Solution 16

Question 18

Find the quadratic polynomial whose zeros are . Verify the relation between the coefficients and the zeros of the polynomial.Solution 18

Question 20

If x =and x = -3 are the roots of the quadratic equation ax² + 7x + b = 0 then find the values of a and b.Solution 20

Question 21

One zero of the polynomial 3x³ + 16x² + 15x – 18 is Find the other zeros of the polynomial.Solution 21

Exercise Ex. 2B

Question 1

Verify that 3, -2, 1 are the zeros of the cubic polynomial p(x) = x³ – 2x² – 5x + 6 and verify the relation between its zeros and coefficients.Solution 1

Question 2

Verify that  are the zeros of the cubic polynomial p(x) = 3x³ – 10x² – 27x + 10 and verify the relation between its zeros and coefficients.Solution 2

Question 3

Find a cubic polynomial whose zeros are 2, -3 and 4.Solution 3

Question 4

Find a cubic polynomial whose zeros are 1 and -3.Solution 4

Question 5

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and the product of its zeros as 5, -2 and -24 respectively.Solution 5

Question 6

Find the quotient and the remainder when:

f(x) = x³ – 3x² + 5x – 3 is divided by g(x)= x² – 2.Solution 6

Question 7

Find the quotient and the remainder when:

f(x)= x⁴ -3x² + 4x + 5 is divided by g(x)= x² + 1 – x.Solution 7

Question 8

Find the quotient and the remainder when:

f(x)= x⁴ – 5x + 6 is divided by g(x) = 2 – x².Solution 8

Question 9

By actual division, show that x² – 3 is a factor of 2x⁴ + 3x³ – 2x² – 9x – 12.Solution 9

Question 11

On dividing 3x³ + x² + 2x + 5 by a polynomial g(x), the quotient and remainder are (3x – 5) and (9x + 10) respectively. Find g(x).

Solution 11

Question 12

Verify division algorithm for the polynomials f(x) = 8 + 20x + x² – 6x³ and g(x) = 2 + 5x – 3x².Solution 12

Question 13

It is given that -1 is one of the zeros of the polynomial x³ + 2x² – 11x – 12. Find all the zeros of the given polynomial.Solution 13

1
Question 14

If 1 and -2 are two zeros of the polynomial, find its third zero.Solution 14

Question 15

If 3 and -3 are two zeros of the polynomial, find all the zeroes of the given polynomial.
Solution 15

Question 16

If 2 and -2 are two zeros of the polynomial, find all the zeros of the given Polynomial.
Solution 16

Question 17

Find all the zeros of, if it is given that two of its zeros are Solution 17

Question 18

Obtain all other zeros of , if two of its zeros are .Solution 18

Question 23

Find all the zeros of the polynomial , it being given that two of its zeros are .Solution 23

Exercise Ex. 2C

Question 1

If one zero of the polynomial x² – 4x + 1 is (2 +), write the other zero.Solution 1

Question 2

Find the zeros of the polynomial x² + x – p(p + 1).Solution 2

Question 3

Find the zeros of the polynomial x² – 3x – m(m + 3).Solution 3

Question 4

Solution 4

Question 5

If one zero of the quadratic polynomial kx² + 3x + k is 2 then find the value of k.Solution 5

Question 6

If 3 is a zero of the polynomial 2x² + x + k, find the value of k.Solution 6

Question 7

If -4 is a zero of the polynomial x² – x – (2k + 2) then find the value of k.Solution 7

Question 8

If 1 is a zero of the polynomial ax² – 3(a – 1)x – 1 then find the value of a.Solution 8

Question 9

If -2 is a zero of the polynomial 3x² + 4x + 2k then find the value of k.Solution 9

Question 10

Write the zeros of the polynomial x² – x – 6.Solution 10

Question 11

If the sum of the zeros of the quadratic polynomial kx² – 3x + 5 is 1, write the value of k.Solution 11

Question 12

If the product of the zeros of the quadratic polynomial x² – 4x + k is 3 then write the value of k.Solution 12

Question 13

If (x + a) is a factor of (2x² + 2ax + 5x + 10), find the value of a.Solution 13

Question 14

If (a – b), a and (a + b) are zeros of the polynomial 2x³ – 6x² + 5x – 7, write the value of a.Solution 14

Question 15

If x³ + x² – ax + b is divisible by (x² – x), write the values of a and b.Solution 15

Question 16

Solution 16

Question 17

State division algorithm for polynomials.Solution 17

If f(x) and g(x) are any two polynomials with g(x) ≠ 0, then we can always find polynomials q(x) and r(x) such that f(x) = q(x)g(x) + r(x),

where r(x) = 0 or degree r(x) < degree g(x).Question 18

The sum of the zeros and the product of zeros of a quadratic polynomial are and -3 respectively. Write the polynomial.Solution 18

Question 19

Write the zeros of the quadratic polynomial f(x) = 6x² – 3.Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

If the zeros of the polynomial f(x) = x³ – 3x² + x + 1 are (a – b), a and (a + b), find a and b.Solution 25

Exercise MCQ

Question 1

Which of the following is a polynomial?

Solution 1

Correct answer: (d)

An expression of the form p(x) = a₀ + a₁x + a₂x² + ….. + a_nxⁿ, where a_n ≠ 0, is called a polynomial in x of degree n.

Here, a₀, a₁, a₂, ……, a_n are real numbers and each power of x is a non-negative integer.

Question 2

Which of the following is not a polynomial?

Solution 2

Correct answer: (d)

An expression of the form p(x) = a₀ + a₁x + a₂x² + ….. + a_nxⁿ, where a_n ≠ 0, is called a polynomial in x of degree n.

Here, a₀, a₁, a₂, ……, a_n are real numbers and each power of x is a non-negative integer.

Question 3

The zeros of the polynomial x² – 2x – 3 are

(a)-3, 1

(b)-3, -1

(c) 3, -1

(d) 3, 1Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

The sum and the product of the zeros of a quadratic polynomial are 3 and -10 respectively. The quadratic polynomial is

(a) x² – 3x + 10

(b) x² + 3x – 10

(c) x² – 3x – 10

(d) x² + 3x + 10Solution 8

Question 9

A quadratic polynomial whose zeros are 5 and -3, is

(a) x² + 2x – 15

(b) x² – 2x + 15

(c) x² – 2x – 15

(d)none of theseSolution 9

Question 10

(a) 10x² +x + 3

(b) 10x² + x – 3

(c) 10x² – x + 3

(d) 10x² – x – 3Solution 10

Question 11

The zeros of the quadratic polynomial x² + 88x + 125 are

(a) both positive

(b) both negative

(c) one positive and one negative

(d) both equalSolution 11

Question 12

If 𝛼 and 𝛽 are the zeroes of x² + 5x + 8 then the value of (𝛼 + 𝛽) is

(a) 5

(b) -5

(c) 8

(d) -8Solution 12

Question 13

If 𝛼 and 𝛽 are the zeros of 2x² + 5x – 9 then the value of 𝛼𝛽 is

Solution 13

Question 14

If one zero of the quadratic polynomial kx² + 3x + k is 2 then the value of k is

Solution 14

Question 15

If one zero of the quadratic polynomial (k – 1)x² + kx + 1 is -4, then the value of k is

Solution 15

Question 16

If -2 and 3 are the zeros of the quadratic polynomial x² + (a + 1)x + b then

(a) a = -2, b = 6

(b) a = 2, b = -6

(c) a = -2, b = -6

(d) a = 2, b = 6Solution 16

Question 17

If one zero of 3x² + 8x + k be the reciprocal of the other then k = ?

(a) 3

(b) -3

(c) 

(d)Solution 17

Question 18

If the sum of the zeros of the quadratic polynomial kx² + 2x + 3k is equal to the product of its zeros then k = ?

Solution 18

Question 19

(a) 3

(b) -3

(c) 12

(d)-12Solution 19

Question 20

If 𝛼, 𝛽, 𝛾 are the zeros of the polynomial x³ – 6x² – x + 30, then (𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼) = ?

(a) -1

(b) 1

(c) -5

(d)30Solution 20

Question 21

If 𝛼, 𝛽, 𝛾 be the zeros of the polynomial 2x³ + x² – 13x + 6, then 𝛼𝛽𝛾 

(a) -3

(b) 3

(c) 

(d)  Solution 21

Question 22

If 𝛼, 𝛽, 𝛾 be the zeros of the polynomial p(x) such that (𝛼 + 𝛽 + 𝛾) = 3, (𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼) = -10 and 𝛼𝛽𝛾 = -24, then p(x) =?

(a) x³ + 3x² – 10x + 24

(b) x³ + 3x² + 10x – 24

(c) x³ – 3x² – 10x + 24

(d) None of theseSolution 22

Question 23

If two of the zeros of the cubic polynomial ax³ + bx² + cx + d are 0, then the third zero is

Solution 23

Question 24

If one of the zeros of the cubic polynomial ax³ + bx² + cx + d is 0, then the product of other two zeros are

Solution 24

Question 25

If one of the zeros of the cubic polynomial x³ + ax² + bx + c is -1, then the product of the other two zeros is

(a) a – b – 1

(b) b – a – 1

(c) 1 – a + b

(d) 1 + a – bSolution 25

Question 26

(a) 3

(b) -3

(c) -2

(d) 2Solution 26

Question 27

On dividing a polynomial p(x) by a non-zero polynomial q(x), let g(x) be the quotient and r(x) be the remainder, then p(x) = q(x).g(x) + r(x), where

(a)r(x) = 0 always

(b)deg r(x) < deg g(x) always

(c) either r(x) = 0 or deg r(x) < deg g(x)

(d) r(x) = g(x)Solution 27

Question 28

Which of the following is a true statement?

(a)x² + 5x – 3 is a linear polynomial

(b)x² + 4x – 1 is a binomial

(c) x + 1 is a monomial

(d) 5x³ is a monomialSolution 28

Exercise FA

Question 1

Zeros of p(x) = x² – 2x – 3 are

(a) 1, -3

(b) 3, -1

(c) -3, -1

(d)1, 3Solution 1

Question 2

If 𝛼, 𝛽, 𝛾 are the zeros of the polynomial x³ – 6x² – x + 30, then (𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼) = ?

(a) -1

(b) 1

(c) -5

(d)30Solution 2

Question 3

If 𝛼, 𝛽 are the zeros of kx² – 2x + 3k such that 𝛼 + 𝛽 = 𝛼𝛽, then k = ?

Solution 3

Question 4

It is given that the difference between the zeros of 4x² – 8kx + 9 is 4 and k > 0. Then, k = ?

Solution 4

Question 5

Find the zeros of the polynomial x² + 2x – 195.Solution 5

Question 6

If one zeros of the polynomial (a² + 9)x² + 13x + 6a is the reciprocal of the other, find the value of a.Solution 6

Question 7

Find a quadratic polynomial whose zeros are 2 and -5.Solution 7

Question 8

If the zeros of the polynomial x³ – 3x² + x + 1 are (a – b), a and (a + b), find the values of a and b.Solution 8

Question 9

Verify that 2 is a zero of the polynomial x³ + 4x² – 3x – 18.Solution 9

Question 10

Find the quadratic polynomial, the sum of whose zeros is -5 and their product is 6.Solution 10

Question 11

Find a cubic polynomial whose zeros are 3, 5 and -2.Solution 11

Question 12

Using remainder theorem, find the remainder when p(x) = x³ + 3x² – 5x + 4 is divided by (x – 2).Solution 12

Question 13

Show that (x + 2) is a factor of f(x) = x³ + 4x² + x – 6.Solution 13

Question 14

Solution 14

Question 15

If 𝛼, 𝛽 are the zeros of the polynomial f(x) = x² – 5x + k such that 𝛼 – 𝛽 = 1, find the value of k.Solution 15

Question 16

Show that the polynomial f(x) = x⁴ + 4x² + 6 has no zero.Solution 16

Question 17

If one zero of the polynomial p(x) = x³ – 6x² + 11x – 6 is 3, find the other two zeros.Solution 17

Question 18

Solution 18

Question 19

Find the quotient when p(x) = 3x⁴ + 5x³ – 7x² + 2x + 2 is divided by (x² + 3x + 1).Solution 19

Question 20

Use remainder theorem to find the value of k, it being given that when x³ + 2x² + kx + 3 is divided by (x – 3), then the remainder is 21.Solution 20