Solution of Chapter 2. Polynomials (RS Aggarwal - Mathematics Book)

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

x² + 7x + 12

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

x² + 2x – 8

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

x² + 3x – 10

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

4x² – 4x – 3

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

5x² – 4 – 8x

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

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

2x² – 11x + 15

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

4x² – 4x + 1

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

x² – 5

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

8x² – 4

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

5y² + 10y

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

3x² – x – 4

Find the quadratic polynomial whose zeros are 2 and ‒6. Verify the relation between the coefficients and the zeros of the polynomial.

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

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

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

Find the quadratic polynomial, the sum of whose roots is √2 and their product is 1/3.

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

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

One zero of the polynomial 3x³ + 16x² + 15x – 18 is 2/3. Find the other zeros of the polynomial.

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.

Verify that 5, –2 and are the zeros of the cubic polynomial p(x) = 3x³ - 10x²– 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) = x³ – 3x² + 5x –3 is divided by g(x) = x² – 2.

Find the quotient and the remainder when:

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

Find the quotient and the remainder when:

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

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

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).

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

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.

If 1 and ‒2 are two zeros of the polynomial (x³ ‒ 4x² ‒ 7x + 10), find its third zero.

If 3 and ‒3 are two zeros of the polynomial (x⁴ + x³ ‒ 11x² ‒ 9x + 18), find all the zeros of the given polynomial.

If 2 and ‒2 are two zeros of the polynomial (X⁴ + x³ ‒ 34x² ‒ 4x + 120), find all the zeros of the given polynomial.

Find all the zeros of (x⁴ + x³ ‒ 23x² ‒ 3x + 60), if it is given that two of its zeros are √3 and ‒√3.

Find all the zeros of (2x⁴ ‒ 3x³ ‒ 5x² + 9x ‒ 3), it being given that two of its zeros are √3 and – √3.

Obtain all other zeros of (x⁴ + 4x³ ‒ 2x² ‒ 20x ‒15) if two of its zeros are √5 and – √5.

Find all the zeros of the polynomial (2x⁴ ‒ 11x³ + 7x² + 13x ‒ 7), it being given that two of its zeros are (3 + √3) and (3 ‒ √3)

If one zero of the polynomial x² ‒ 4x + 1 is (2 + √3), write the other zero.

Find the zeros of the polynomial x² + x ‒ p (p + 1).

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

Find α, β are the zeros of a polynomial such that α + β= 6 and αβ = 4 then write the polynomial.

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

If 3 is a zero of the polynomial 2s₂ + x + k, find the value of k.

If ‒4 is a zero of the polynomial x² ‒ x (2k + 2) then find the value of k.

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

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

Write the zeros of the polynomial x² ‒ x ‒ 6.

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

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

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

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

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

If α and β are the zeros of the polynomial 2x² + 7x + 5, write the value of α + β + αβ.

State division algorithm for polynomials.

The sum of the zeros and the product of zeros of a quadratic polynomial are -1/2 and ‒3 respectively. Write the polynomial.

Write the zeros of the quadratic polynomial f(X) = 6x² ‒ 3.

Write the zeros of the quadratic polynomial f(x) = 4√3x² + 5x ‒ 2√3.

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

If α and β are the zeros of the polynomial f(x) = 6x² + x ‒ 2, find the value of

If α and β are the zeros of the polynomial f(x) = 5x² ‒ 7x + 1, find the value of

If α and β are the zeros of the polynomial f(x) = x² + x – 2, find the value of

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

Which of the following is a polynomial?

Which of the following is not a polynomial?

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

The zeros of the polynomial x² ‒ √2 x – 12 are

The zeros of the polynomial 4x² + 5√2x – 3 are

The zeros of the polynomial are

The zeros of the polynomial are

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

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

A quadratic polynomial whose zeros are , is

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

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

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

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

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

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

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

If the sum of the zeros of the quadratic polynomial kx² + 2x + 3k is equal

If α, β are the zeros of the polynomial x² + 6x + 2 then ?

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

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

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

If two of the zeros of the cubic polynomial az³ + bx² + cx + d are 0 then in the third zero is

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

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

If α, β be the zeros of the polynomial 2x² + 5x + k such that then k = ?

One dividing a polynomial p(x) by a nonzero 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

Which of the following is a true statement?

Zeros of p(x) = x² – 2x – 3 are

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

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

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

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

If one zero of the polynomial (a² + 9) x² + 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 x³ – 3x² + 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 x³ + 4x²– 3x – 18.

Find the quadratic polynomial, the sum of whose zeros is ‒5 and their products is 6.

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

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

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

If α, β, γ are the zeros of the polynomial p(x) = 6x³ + 3x² – 5x + 1, find the value of .

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

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

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

If two zeros of the polynomial p(x) = 2x⁴ – 3x³ – 3x² + 6x – 2 are √2 and – √2, find its other two zeros.

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

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.