1

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

x^{2} + 7x + 12

2

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

x^{2} + 2x – 8

3

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

x^{2} + 3x – 10

4

4x^{2} – 4x – 3

5

5x^{2} – 4 – 8x

6

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7

2x^{2} – 11x + 15

8

4x^{2} – 4x + 1

9

x^{2} – 5

10

8x^{2} – 4

11

5y^{2} + 10y

12

3x^{2} – x – 4

13

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

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14

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

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15

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

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16

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

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17

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

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18

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

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19

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

20

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

21

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

1

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.

2

Verify that 5, –2 and are the zeros of the cubic polynomial p(x) = 3x^{3} - 10x^{2}– 27x + 10 and verify the relation between its zeros and coefficients

3

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

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4

Find a cubic polynomial whose zeros are , 1 and –3.

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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.

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6

Find the quotient and the remainder when:

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

7

Find the quotient and the remainder when:

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

8

Find the quotient and the remainder when:

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

9

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

10

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).

11

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

12

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.

13

If 1 and ‒2 are two zeros of the polynomial (x^{3} ‒ 4x^{2} ‒ 7x + 10), find its third zero.

14

If 3 and ‒3 are two zeros of the polynomial (x^{4} + x^{3} ‒ 11x^{2} ‒ 9x + 18), find all the zeros of the given polynomial.

15

If 2 and ‒2 are two zeros of the polynomial (X^{4} + x^{3} ‒ 34x^{2} ‒ 4x + 120), find all the zeros of the given polynomial.

16

Find all the zeros of (x^{4} + x^{3} ‒ 23x^{2} ‒ 3x + 60), if it is given that two of its zeros are √3 and ‒√3.

17

Find all the zeros of (2x^{4} ‒ 3x^{3} ‒ 5x^{2} + 9x ‒ 3), it being given that two of its zeros are √3 and – √3.

18

Obtain all other zeros of (x^{4} + 4x^{3} ‒ 2x^{2} ‒ 20x ‒15) if two of its zeros are √5 and – √5.

19

Find all the zeros of the polynomial (2x^{4} ‒ 11x^{3} + 7x^{2} + 13x ‒ 7), it being given that two of its zeros are (3 + √3) and (3 ‒ √3)

1

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

2

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

3

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

4

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

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5

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

6

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

7

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

8

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

9

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

10

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

11

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

12

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

13

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

14

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

15

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

16

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

17

State division algorithm for polynomials.

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18

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

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19

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

20

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

21

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

22

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

23

If α and β are the zeros of the polynomial f(x) = 5x^{2} ‒ 7x + 1, find the value of

24

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

25

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

1

Which of the following is a polynomial?

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2

Which of the following is not a polynomial?

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3

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

4

The zeros of the polynomial x^{2} ‒ √2 x – 12 are

5

The zeros of the polynomial 4x^{2} + 5√2x – 3 are

6

The zeros of the polynomial are

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7

The zeros of the polynomial are

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8

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

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9

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

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10

A quadratic polynomial whose zeros are , is

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11

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

12

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

13

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

14

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

15

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

16

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

17

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

18

If the sum of the zeros of the quadratic polynomial kx^{2} + 2x + 3k is equal

19

If α, β are the zeros of the polynomial x^{2} + 6x + 2 then ?

20

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

21

If α, β, γ are the zeros of the polynomial 2x^{3} + x^{2} – 13x + 6 then αβγ = ?

22

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) = ?

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23

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

24

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

25

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

26

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

27

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

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28

Which of the following is a true statement?

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1

Zeros of p(x) = x^{2} – 2x – 3 are

2

If α, β, γ are the zeros of the polynomial x^{3} – 6x^{2} – x + 30 then the value of (αβ + βγ + γ α) is

3

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

4

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

5

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

6

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

7

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

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8

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

9

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

10

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

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11

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

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12

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

13

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

14

If α, β, γ are the zeros of the polynomial p(x) = 6x^{3} + 3x^{2} – 5x + 1, find the value of .

15

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.

16

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

17

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

18

If two zeros of the polynomial p(x) = 2x^{4} – 3x^{3} – 3x^{2} + 6x – 2 are √2 and – √2, find its other two zeros.

19

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

20

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.