R.D. Sharma solutions for मैथमैटिक्स [अंग्रेजी] कक्षा १० chapter 2 - Polynomials [Latest edition]

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

f(x) = x² – 2x – 8

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

g(s) = 4s² – 4s + 1

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

h(t) = t² – 15

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

f(x) = 6x² – 3 – 7x

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

`q(y) = 7y^2 - 11/3y - 2/3`

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

`phi(x) = 2x^2 + 7/2x + 3/4`

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

`p(x) = x^2 + 4/3x - 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 factorization.

`-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 factorization.

`21/8, 5/16`

If α and β are the zeros of the quadratic polynomial f(x) = x² – 5x + 4, find the value of `1/α + 1/β - 2αβ`.

If α and β are the zeros of the quadratic polynomial p(y) = 5y² – 7y + 1, find the value of `1/α + 1/β`.

If one zero of the quadratic polynomial f(x) = 4x² – 8kx – 9 is negative of the other, find the value of k.

If the sum of the zeros of the quadratic polynomial f(t) = kt² + 2t + 3k is equal to their product, find the value of k.

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

`p(x) = x^2 + 2sqrt(2)x - 6`

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

`q(x) = sqrt(3)x^2 + 10x + 7sqrt(3)`

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

`f(x) = x^2 - (sqrt(3) + 1)x + sqrt(3)`

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

g(x) = a(x² + 1) – x(a² + 1)

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

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

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

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

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

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

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

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

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

f(x) = 4x² + 4x – 3

Find a quadratic polynomial whose sum and product respectively of the zeros are given. Also, find the zeros of these polynomials.

`-2sqrt(3), -9`

Find a quadratic polynomial whose sum and product respectively of the zeros are given. Also, find the zeros of these polynomials.

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

If α and β are the zeros of the quadratic polynomial p(x) = 4x² – 5x – 1, find the value of α²β + αβ².

If α and β are the zeros of the quadratic polynomial f(x) = 6x² + x – 2, find the value of `alpha/beta + beta/alpha`.

If α and β are the zeros of the quadratic polynomial f(x) = x² – p(x + 1) – c, show that (α + 1)(β + 1) = 1 – c.

If α and β are the zeros of a quadratic polynomial such that α + β = 24 and α – β = 8, find a quadratic polynomial having α and β as its zeros.

Find a quadratic polynomial whose zeros are negative of the zeros of the polynomial px² + qx + r.

If α and β are zeroes of a quadratic polynomial px² + qx + 1. Form a quadratic polynomial whose zeroes are `2/α` and `2/β`.

If α and β are the zeroes of the polynomial ax² – x + c. Obtain a polynomial whose zeroes are α – 3 and β – 3.

Obtain the zeroes of the polynomial 7x² + 18x – 9. Hence, write a polynomial each of whose zeroes is twice the zeroes of the given polynomial.

Obtain the zeroes of the polynomial p(x) = 2x² – 5x – 3. Hence, obtain a polynomial each of whose zeroes is one less than each of the zero of p(x).

Find the zeroes of the polynomial p(x) = 3x² – 4x – 4. Hence, write a polynomial whose each of the zero is 2 more than zeroes of p(x).

Find the zeroes of the polynomial q(x) = 8x² – 2x – 3. Hence, find a polynomial whose zeroes are 2 less than the zeroes of q(x).

Find the zeroes of the polynomial r(x) = 4x² + 3x – 1. Hence, write a polynomial whose zeroes are reciprocal of the zeroes of polynomial r(x).

If α and β are the zeros of the quadratic polynomial f(x) = x² – 1, find a quadratic polynomial whose zeros are `(2alpha)/beta` and `(2beta)/alpha`.

If the squared difference of the zeros of the quadratic polynomial f(x) = x² + px + 45 is equal to 144, find the value of p.

If α and β are the zeros of the quadratic polynomial f(x) = x² – px + q, prove that `alpha^2/beta^2 + beta^2/alpha^2 = p^4/q^2 - (4p^2)/q + 2`.

If α and β are the zeroes of the quadratic polynomial f(x) = ax² + bx + c, then evaluate `1/(aalpha + b) + 1/(abeta + b)`.

If α and β are the zeros of the quadratic polynomial f(x) = ax² + bx + c, then evaluate `beta/(aalpha + b) + alpha/(abeta + b)`.

If α and β are the zeros of the quadratic polynomial f(x) = ax² + bx + c, then evaluate `a(α^2/β + β^2/α) + b(α/β + β/α)`.

If a and b are the zeros of the quadratic polynomial f(x) = x² + x – 2, find the value of `α/β + β/α`.

If a and b are the zeros of the quadratic polynomial f(x) = x² + x – 2, find the value of `1/α - 1/β`.

Verify that the numbers given along side of the cubic polynomials are their zeroes. Also, verify the relationship between the zeros and the coefficients.

`f(x) = 2x^3 + x^2 - 5x + 2 ; 1/2, 1, - 2`

Verify that the numbers given along side of the cubic polynomials below are their zeroes. Also verify the relationship between the zeros and the coefficients.

g(x) = x³ – 4x² + 5x – 2; 2, 1, 1

Find a cubic polynomial with the sum, sum of the product of its zeros taken two at a time, and product of its zeros as 3, –1 and –3 respectively.

If the zeros of the polynomial f(x) = 2x³ – 15x² + 37x – 30 are in A.P., find them.

Find the condition that the zeros of the polynomial f(x) = x³ + 3px² + 3qx + r may be in A.P.

If the zeros of the polynomial f(x) = ax³ + 3bx² + 3cx + d are in A.P., prove that 2b³ – 3abc + a²d = 0.

If the zeros of the polynomial f(x) = x³ – 12x² + 39x + k are in A.P., find the value of k.

If 4 is a zero of the cubic polynomial x³ – 3x² – 10x + 24, find its other two zeros.

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

t² – 3, 2t⁴ + 3t³ – 2t² – 9t – 12

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

x³ – 3x + 1, x⁵ – 4x³ + x² + 3x + 1

Check whether the first polynomial is a factor of the second polynomial by applying the division algorithm:

g(x) = 2x² – x + 3, f(x) = 6x⁵ – x⁴ + 4x³ – 5x² – x – 15

Obtain all zeros of f(x) = x³ + 13x² + 32x + 20, if one of its zeros is –2.

Obtain all zeros of the polynomial f(x) = 2x⁴ + x³ – 14x² – 19x – 6, if two of its zeros are –2 and –1.

Find all the zeros of the polynomial x⁴ + x³ – 34x² – 4x + 120, if two of its zeros are 2 and –2.

For what value of k, is the polynomial f(x) = 3x⁴ – 9x³ + x² + 15x + k completely divisible by 3x² – 5?

Find all the zeros of the polynomial 2x³ + x² – 6x – 3, if two of its zeros are `-sqrt(3)` and `sqrt(3)`.

Find all the zeros of the polynomial x³ + 3x² – 2x – 6, if two of its zeros are `-sqrt(2)` and `sqrt(2)`.

Find all zeros of the polynomial f(x) = 2x⁴ – 2x³ – 7x² + 3x + 6, if its two zeros are `-sqrt(3/2)` and `sqrt(3/2)`.

Find all zeros of the polynomial 2x⁴ + 7x³ – 19x² – 14x + 30, if two of its zeros are `sqrt(2)` and `-sqrt(2)`.

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.

What must be added to the polynomial f(x) = x⁴ + 2x³ – 2x² + x – 1 so that the resulting polynomial is exactly divisible by x² + 2x – 3?

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following:

f(x) = x³ – 6x² + 11x – 6, g(x) = x² + x + 1

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following:

f(x) = 10x⁴ + 17x³ – 62x² + 30x – 3, g(x) = 2x² + 7x + 1

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following:

f(x) = 4x³ + 8x + 8x² + 7, g(x) = 2x² – x + 1

Apply division algorithm to find the quotient q(x) and remainder r(x) on dividing f(x) by g(x) in the following:

f(x) = 15x³ – 20x² + 13x – 12, g(x) = 2 – 2x + x²

The sum and product of the zeros of a quadratic polynomial are \[- \frac{1}{2}\] and –3 respectively. What is the quadratic polynomial.

Write the family of quadratic polynomials having \[- \frac{1}{4}\] and 1 as its zeros.

Find a quadratic polynomial whose zeroes are 2 and `-7/5`.

In figure, the graph of a polynomial p(x) is given. Find the zeros of the polynomial.

The graph of a polynomial y = f(x), shown in figure. Find the number of real zeros of f(x).

The graph of a polynomial f(x) is as shown in figure. Write the number of real zeros of f(x).

Write a quadratic polynomial, sum of whose zeros is \[2\sqrt{3}\] and their product is 2.

If α, β are zeroes of the polynomial p(x) = 5x² – 6x + 1, then find the value of α + β + αβ.

If α and β are zeroes of the polynomial p(x) = x² – 2x – 1, then find the value of `1/(2α) + 1/(2β) + 3αβ`.

If the sum of the zeroes of the polynomial p(x) = (p + 1)x² + (2p + 3)x + (3p + 4) is –1, then find the value of p.

Find a quadratic polynomial whose zeroes are 2 and `-7/5`.

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

The graph of the polynomial f(x) = ax² + bx + c is as shown below. Write the signs of 'a' and b² – 4ac.

The graph of the polynomial f(x) = ax² + bx + c is as shown in figure. Write the value of b² – 4ac and the number of real zeros of f(x).

In Q. No. 36, write the sign of c.

In Q. No. 37, write the sign of c.

Quadratic polynomials, whose zeroes are – 4 and 3 are given by ______.

The number of quadratic polynomials whose zeroes are 2 and – 3, is  ______.

The zeroes of the quadratic polynomial x² + x – 6 are ______.

The quadratic polynomials, the sum and product of whose zeroes are 7 and 12 are given by ______.

If x + 1 is a factor of the polynomial 2x³ + ax² + 4x + 1, then a = ______.

The graph of a quadratic polynomial intersects the x-axis at the most at ______ points.

If – 4 is a zero of the polynomial x² – x – (2k + 2), then k = ______.

If 4x² – 6x – m is divisible by x – 3, then m = ______.

If one zero of the quadratic polynomial 2x² – 6kx + 6x – 7 is negative of the other, then k = ______.

The sum of the zeros of the quadratic polynomial 2x² – 3k is ______.

Quadratic polynomials with rational coefficients having `sqrt(3)` as a zero are given by ______.

If x + α is a factor of the polynomial 2x² + 2αx + 4x + 12, then α = ______.

If the product of the zeroes of the quadratic polynomial x² – 3ax + 2a² – 1 is 7, then a = ______.

The maximum number of zeros which a quadratic polynomial can have is ______.

If a + b + c = 0, then a zero of the polynomial ax² + bx + c, is ______.

If a + c = b, then a zero of the polynomial ax² + bx + c, is ______.

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

If all the zeroes of cubic polynomial x³ + ax² + bx + c are negative then a, b and c all have ______ sign.

If the zeroes of the quadratic polynomial ax² + x + a are equal, then a = ______.

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

If a, a + b, a + 2b are zeroes of the cubic polynomial x³ – 6x² + 3x + 10, then a + b = ______.

The parabola representing a quadratic polynomial f(x) = ax² + bx + c opens upward when ______.

The parabola representing a quadratic polynomial f(x) = ax² + bx + c opens downward when ______.

If the parabola represented by f(x) = ax² + bx + c cuts x-axis at two distinct points, then the polynomial ax² + bx + c has ______ real zeroes.