NCERT Exemplar solutions for Mathematics [English] Class 10 chapter 2 - Polynomials [Latest edition]

If one of the zeroes of the quadratic polynomial (k – 1)x² + k x + 1 is –3, then the value of k is ______.

A quadratic polynomial, whose zeroes are –3 and 4, is ______.

If the zeroes of the quadratic polynomial x² + (a + 1) x + b are 2 and –3, then ______.

The number of polynomials having zeroes as –2 and 5 is ______.

Given that one of the zeroes of the cubic polynomial ax³ + bx² + cx + d is zero, the product of the other two zeroes is ______.

If one of the zeroes of the cubic polynomial x³ + ax² + bx + c is –1, then the product of the other two zeroes is ______.

The zeroes of the quadratic polynomial x² + 99x + 127 are ______.

The zeroes of the quadratic polynomial x² + kx + k, k ≠ 0 ______.

If the zeroes of the quadratic polynomial ax² + bx + c, c ≠ 0 are equal, then ______.

If one of the zeroes of a quadratic polynomial of the form x² + ax + b is the negative of the other, then it ______.

Which of the following is not the graph of a quadratic polynomial?

Can x² – 1 be the quotient on division of x⁶ + 2x³ + x – 1 by a polynomial in x of degree 5?

What will the quotient and remainder be on division of ax² + bx + c by px³ + qx² + rx + s, p ≠ 0?

If on division of a polynomial p(x) by a polynomial g(x), the quotient is zero, what is the relation between the degrees of p(x) and g(x)?

If on division of a non-zero polynomial p(x) by a polynomial g(x), the remainder is zero, what is the relation between the degrees of p(x) and g(x)?

Can the quadratic polynomial x² + kx + k have equal zeroes for some odd integer k > 1?

If the zeroes of a quadratic polynomial ax² + bx + c are both positive, then a, b and c all have the same sign.

If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial.

If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial.

If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms.

If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.

If all three zeroes of a cubic polynomial x³ + ax² – bx + c are positive, then at least one of a, b and c is non-negative.

The only value of k for which the quadratic polynomial kx² + x + k has equal zeros is `1/2`

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

4x² – 3x – 1

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

3x² + 4x – 4

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

5t² + 12t + 7

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

t³ – 2t² – 15t

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

`2x^2 + (7/2)x + 3/4`

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

`4x^2 + 5sqrt(2)x - 3`

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

`2s^2 - (1 + 2sqrt(2))s + sqrt(2)`

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

`v^2 + 4sqrt(3)v - 15`

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

`y^2 + 3/2 sqrt(5)y - 5`

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

`7y^2 - 11/3 y - 2/3`

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.

`(-8)/3, 4/3`

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.

`21/8, 5/16`

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.

`-2sqrt(3), -9`

For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.

`(-3)/(2sqrt(5)), -1/2`

Given that the zeroes of the cubic polynomial x³ – 6x² + 3x + 10 are of the form a, a + b, a + 2b for some real numbers a and b, find the values of a and b as well as the zeroes of the given polynomial.

Given that `sqrt(2)` is a zero of the cubic polynomial `6x^3 + sqrt(2)x^2 - 10x - 4sqrt(2)`, find its other two zeroes.

Find k so that x² + 2x + k is a factor of 2x⁴ + x³ – 14 x² + 5x + 6. Also find all the zeroes of the two polynomials.

Given that `x - sqrt(5)` is a factor of the cubic polynomial `x^3 - 3sqrt(5)x^2 + 13x - 3sqrt(5)`, find all the zeroes of the polynomial.

For which values of a and b, are the zeroes of q(x) = x³ + 2x² + a also the zeroes of the polynomial p(x) = x⁵ – x⁴ – 4x³ + 3x² + 3x + b? Which zeroes of p(x) are not the zeroes of q(x)?