# R S AGGARWAL AND V AGGARWAL Solutions for Class 10 Maths Chapter 2 - Polynomials

## Chapter 2 - Polynomials Exercise Ex. 2A

Find the zeros of the quadratic polynomial (x^{2} + 3x - 10) and verify the relation between its zeros and coefficients.

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

x^{2}
- 2x - 8

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

x^{2} + 7x + 12

Find the zeros of the quadratic polynomial (4x^{2} - 4x - 3) and verify the relation between its zeros and coefficients.

We have

Find the zeros of the quadratic polynomial (5x^{2} - 4 - 8x) and verify the relationship between its zeros and coefficients of the given polynomial.

Find the zeros of the quadratic polynomial (2x^{2} - 11x + 15) and verify the relation between its zeros and coefficients.

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

4x^{2} - 4x + 1

3x^{2} - x - 4

5y^{2}
+ 10y

Find the zeros of the quadratic polynomial (8x^{2} - 4) and verify the relation between its zeros and coefficients.

Let

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

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

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

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

If x =and x = -3 are the roots of the quadratic equation ax^{2} + 7x + b = 0 then find the values of a and b.

One zero of the polynomial 3x^{3} + 16x^{2 }+ 15x - 18 is Find the other zeros of the polynomial.

## Chapter 2 - Polynomials Exercise Ex. 2B

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

Verify that are the zeros of the cubic polynomial p(x) = 3x^{3} - 10x^{2} - 27x + 10 and verify the relation between its zeros and coefficients.

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

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

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.

Find the quotient and the remainder when:

f(x) = x^{3} - 3x^{2}
+ 5x - 3 is divided by g(x)= x^{2} - 2.

Find the quotient and the remainder when:

f(x)=
x^{4} -3x^{2} + 4x + 5 is divided by g(x)= x^{2} + 1 -
x.

Find the quotient and the remainder when:

f(x)= x^{4} - 5x + 6 is
divided by g(x) = 2 - x^{2}.

By
actual division, show that x^{2} - 3 is a factor of 2x^{4} +
3x^{3} - 2x^{2} - 9x - 12.

On dividing 3x^{3} + x^{2} + 2x + 5 by a polynomial g(x), the quotient and remainder are (3x - 5) and (9x + 10) respectively. Find g(x).

Verify division algorithm for the polynomials f(x) = 8 + 20x + x^{2} - 6x^{3} and g(x) = 2 + 5x - 3x^{2}.

It is given that -1 is one of the zeros of the polynomial x^{3} + 2x^{2} - 11x - 12. Find all the zeros of the given polynomial.

1

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

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

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

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

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

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

## Chapter 2 - Polynomials Exercise Ex. 2C

If
one zero of the polynomial x^{2} - 4x + 1 is (2 +), write the other zero.

Find
the zeros of the polynomial x^{2} + x - p(p
+ 1).

Find
the zeros of the polynomial x^{2} - 3x - m(m
+ 3).

If
one zero of the quadratic polynomial kx^{2} + 3x + k is 2 then find
the value of k.

If
3 is a zero of the polynomial 2x^{2} + x + k, find the value of k.

If
-4 is a zero of the polynomial x^{2 }- x - (2k + 2) then find the
value of k.

If
1 is a zero of the polynomial ax^{2} - 3(a - 1)x - 1 then find the
value of a.

If
-2 is a zero of the polynomial 3x^{2} + 4x + 2k then find the value
of k.

Write
the zeros of the polynomial x^{2} - x - 6.

If
the sum of the zeros of the quadratic polynomial kx^{2} - 3x + 5 is
1, write the value of k.

If
the product of the zeros of the quadratic polynomial x^{2} - 4x + k
is 3 then write the value of k.

If
(x + a) is a factor of (2x^{2} + 2ax + 5x + 10), find the value of a.

If
(a - b), a and (a + b) are zeros of the polynomial 2x^{3} - 6x^{2}
+ 5x - 7, write the value of a.

If
x^{3} + x^{2} - ax + b is divisible by (x^{2} - x),
write the values of a and b.

State division algorithm for polynomials.

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).

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

Write
the zeros of the quadratic polynomial f(x) = 6x^{2} - 3.

If
the zeros of the polynomial f(x) = x^{3} - 3x^{2} + x + 1 are
(a - b), a and (a + b), find a and b.

## Chapter 2 - Polynomials Exercise MCQ

Which of the following is a polynomial?

Correct answer: (d)

An expression of the form p(x) = a_{0} + a_{1}x + a_{2}x^{2} + ….. + a_{n}x^{n}, where a_{n} ≠ 0, is called a polynomial in x of degree n.

Here, a_{0}, a_{1}, a_{2}, ……, a_{n} are real numbers and each power of x is a non-negative integer.

Which of the following is not a polynomial?

Correct answer: (d)

An expression of the form p(x) = a_{0} + a_{1}x + a_{2}x^{2} + ….. + a_{n}x^{n}, where a_{n} ≠ 0, is called a polynomial in x of degree n.

Here, a_{0}, a_{1}, a_{2}, ……, a_{n} are real numbers and each power of x is a non-negative integer.

The zeros of the polynomial x^{2 }- 2x - 3 are

(a)-3, 1

(b)-3, -1

(c) 3, -1

(d) 3, 1

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

(a) x^{2} - 3x + 10

(b) x^{2} + 3x - 10

(c) x^{2} - 3x - 10

(d) x^{2} + 3x + 10

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

(a) x^{2} + 2x - 15

(b) x^{2} - 2x + 15

(c) x^{2} - 2x - 15

(d)none of these

(a) 10x^{2} +x + 3

(b) 10x^{2} + x - 3

(c) 10x^{2} - x + 3

(d) 10x^{2} - x - 3

The zeros of the quadratic polynomial x^{2 }+ 88x + 125 are

(a) both positive

(b) both negative

(c) one positive and one negative

(d) both equal

If 𝛼 and 𝛽 are the zeroes of x^{2 }+ 5x + 8 then the value of (𝛼 + 𝛽) is

(a) 5

(b) -5

(c) 8

(d) -8

If 𝛼 and 𝛽 are the zeros of 2x^{2 }+ 5x - 9 then the value of 𝛼𝛽 is

If one zero of the quadratic polynomial kx^{2 }+ 3x + k is 2 then the value of k is

If one zero of the quadratic polynomial (k - 1)x^{2 }+ kx + 1 is -4, then the value of k is

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

(a) a = -2, b = 6

(b) a = 2, b = -6

(c) a = -2, b = -6

(d) a = 2, b = 6

If one zero of 3x^{2} + 8x + k be the reciprocal of the other then k = ?

(a) 3

(b) -3

(c)

(d)

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

(a) 3

(b) -3

(c) 12

(d)-12

If 𝛼, 𝛽, 𝛾 are the zeros of the polynomial x^{3 }- 6x^{2 }- x + 30, then (𝛼𝛽 + 𝛽𝛾 + 𝛾𝛼) = ?

(a) -1

(b) 1

(c) -5

(d)30

If 𝛼, 𝛽, 𝛾 be the zeros of the polynomial 2x^{3 }+ x^{2 }- 13x + 6, then 𝛼𝛽𝛾

(a) -3

(b) 3

(c)

(d)

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) x^{3 }+ 3x^{2 }- 10x + 24

(b) x^{3 }+ 3x^{2 }+ 10x - 24

(c) x^{3 }- 3x^{2 }- 10x + 24

(d) None of these

If two of the zeros of the cubic polynomial ax^{3 }+ bx^{2 }+ cx + d are 0, then the third zero is

If one of the zeros of the cubic polynomial ax^{3 }+ bx^{2 }+ cx + d is 0, then the product of other two zeros are

If one of the zeros of the cubic polynomial x^{3 }+ ax^{2 }+ 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 - b

(a) 3

(b) -3

(c) -2

(d) 2

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)

Which of the following is a true statement?

(a)x^{2 }+ 5x - 3 is a linear polynomial

(b)x^{2 }+ 4x - 1 is a binomial

(c) x + 1 is a monomial

(d) 5x^{3} is a monomial

## Chapter 2 - Polynomials Exercise FA

Zeros of p(x) = x^{2 }- 2x - 3 are

(a) 1, -3

(b) 3, -1

(c) -3, -1

(d)1, 3

If 𝛼, 𝛽, 𝛾 are the zeros of the polynomial x^{3 }- 6x^{2
}- x + 30, then (𝛼𝛽
+ 𝛽𝛾 + 𝛾𝛼) = ?

(a) -1

(b) 1

(c) -5

(d)30

If 𝛼, 𝛽 are the zeros of kx^{2 }- 2x + 3k
such that 𝛼
+ 𝛽 = 𝛼𝛽, then k = ?

It is given that the difference between the
zeros of 4x^{2} - 8kx + 9 is 4 and k > 0. Then, k = ?

Find the zeros of the polynomial x^{2 }+
2x - 195.

If one zeros of the polynomial (a^{2 }+
9)x^{2 }+ 13x + 6a is the reciprocal of the other, find the value of
a.

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

If the zeros of the polynomial x^{3 }-
3x^{2 }+ x + 1 are (a - b), a and (a + b), find the values of a and
b.

Verify that 2 is a zero of the polynomial x^{3
}+ 4x^{2 }- 3x - 18.

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

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

Using remainder theorem, find the remainder
when p(x) = x^{3 }+ 3x^{2 }- 5x + 4 is divided by (x - 2).

Show that (x + 2) is a factor of f(x) = x^{3
}+ 4x^{2 }+ x - 6.

If 𝛼, 𝛽 are the zeros of the polynomial f(x) = x^{2
}- 5x + k such that 𝛼
- 𝛽 = 1, find the value of k.

Show that the polynomial f(x) = x^{4 }+
4x^{2 }+ 6 has no zero.

If one zero of the polynomial p(x) = x^{3
}- 6x^{2 }+ 11x - 6 is 3, find the other two zeros.

Find the quotient when p(x) = 3x^{4 }+
5x^{3 }- 7x^{2 }+ 2x + 2 is divided by (x^{2 }+ 3x + 1).

Use remainder theorem to find the value of
k, it being given that when x^{3 }+ 2x^{2 }+ kx + 3 is
divided by (x - 3), then the remainder is 21.

### Other Chapters for CBSE Class 10 Mathematics

Chapter 1- Real Numbers Chapter 3- Linear equations in two variables Chapter 4- Quadratic Equations Chapter 5- Arithmetic Progressions Chapter 6- Co-ordinate Geometry Chapter 7- Trigonometric Ratios of Complementary Angles Chapter 8- Trigonometric Identities Chapter 9- Mean, Median, Mode of Grouped Data, Cumulative Frequency Graph and Ogive Chapter 12- Circles Chapter 13- Constructions Chapter 14- Height and Distance Chapter 15- Probability Chapter 17- Perimeter and Areas of Plane Figures Chapter 18- Areas of Circle, Sector and Segment Chapter 19- Volume and Surface Areas of Solids### R S AGGARWAL AND V AGGARWAL Solutions for CBSE Class 10 Subjects

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