1

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

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

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2

If α and β are the zeros of the quadratic polynomial, find the value of

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3

If α and β are the zeros of the quadratic polynomial, find the value of .

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4

If α and β are the zeros of the quadratic polynomial , find the value of

.

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5

If α and β are the zeros of the quadratic polynomial , find the value of .

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6

If α and β are the zeros of the quadratic polynomial , find the value of .

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7

If α and β are the zeros of the quadratic polynomial , find the value of .

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8

If α and β are the zeros of the quadratic polynomial , find the value of

.

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9

If α and β are the zeros of the quadratic polynomial , find the value of .

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10

If one zero of the quadratic polynomial is negative of the other, find the value of k.

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11

If the sum of the zeros of the quadratic polynomial is equal to their product, find the value of k.

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12

If the squared difference of the zeros of the quadratic polynomial is equal to 144, find the value of p.

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13

If α and β are the zeros of the quadratic polynomial , prove that .

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14

If α and β are the zeros of the quadratic polynomial , show that .

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15

If α and β are the zeros of the quadratic polynomial such that and, find a quadratic polynomial having α and β as its zeros.

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16

If α and β are the zeros of the quadratic polynomial , find a quadratic polynomial whose zeros are .

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17

If α and β are the zeros of the quadratic polynomial , find a quadratic polynomial whose zeros are .

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18

If α and β are the zeros of the quadratic polynomial , from a polynomial whose zeros are .

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19

If α and β are the zeros of the quadratic polynomial , find a polynomial whose roots are (i) (ii) .

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20

If α and β are the zeros of the quadratic polynomial , then evaluate:

(i) (ii)

(iii) (iv)

(v) (vi)

(vii) (viii)

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1

Verify that the numbers given along side of the cubic polynomials below are their zeros. Also, verify the relationship between the zeros and coefficients in each case:

(i)

(ii)

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2

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.

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3

If the zeros of the polynomial are in A.P., find them.

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4

Find the condition that the zeros of the polynomial may be in A.P.

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5

If the zeros of the polynomial are in A.P., prove that .

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6

If the zeros of the polynomial are in A.P., find the value of k.

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1

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

(i)

(ii)

(iii)

(iv)

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2

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

(i)

(ii)

(iii)

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3

Obtain all zeros of the polynomial , if two of its zeros are -2 and -1.

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4

Obtain all zeros of if one of its zeros is -2.

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5

Obtain all zeros of the polynomial if two of its zeros are and .

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6

Find all zeros of the polynomial it its two zeros are and .

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7

What must be added to the polynomial so that the resulting polynomial is exactly divisible by ?

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8

What must be subtracted from the polynomial so that the resulting polynomial is exactly divisible by ?

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9

Find all the zeros of the polynomial , if two of its zeros are 2 and-2.

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10

Find all zeros of the polynomial , if two of its zeros are and .

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11

Find all the zeros of the polynomial , if two of its zeros are and .

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12

Find all the zeros of the polynomial , if two of its zeros are and .

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1

Define a polynomial with real coefficients.

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2

Define degree of a polynomial.

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3

Write the standard form of a linear polynomial with real coefficients.

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4

Write the standard form of a quadratic polynomial with real coefficients.

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5

Write the standard form of a cubic polynomial with real coefficients.

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6

Define the value of a polynomial at a point.

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7

Define zero of a polynomial.

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8

The sum and product of the zeros of a quadratic polynomial are and — 3 respectively. What is the quadratic polynomial?

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9

Write the family of quadratic polynomials having and 1 as its zeros.

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10

If the product of zeros of the quadratic polynomial f(x) = x^{2} — 4x + k is 3, find the value of k.

11

If the sum of the zeros of the quadratic polynomial f (x) = kx^{2} — 3x + 5 is 1, write the value of k.

12

In Fig. 2.17, the graph of a polynomial p(x) is given. Find the zeros of the polynomial.

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13

The graph of a polynomial y = f (x), shown in Fig. 2.18. Find the number of real zeros of f (x).

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14

The graph of the polynomial f(x) = ax^{2} + bx + c is as shown below (Fig. 2.19). Write the signs of 'a' and b^{2} – 4ac.

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15

The graph of the polynomial f(x) = ax^{2} + box + c is as shown in Fig. 2.20. Write the value of b^{2} – 4ac and the number of real zeros of f (x).

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16

In Q. No. 14, write the sign of c.

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17

In Q. No. 15, write the sign of c.

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18

The graph of a polynomial/ (x) is as shown in Fig. 2.21. Write the number of real zeros of f (x).

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19

If x = 1 is a zero of the polynomial f (x) = x^{3} – 2x^{2} + 4x + k, write the value of k.

20

State division algorithm for polynomials.

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21

Give an example of polynomials f(x), g(x), q(x) and r(x) satisfying f(x) = g(x) .q(x)+ r(x), where degree r (x) = 0.

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22

Write a quadratic polynomial, sum of whose zeros is 2√3 and their product is 2.

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23

If fourth degree polynomial is divided by a quadratic polynomial, write the degree of the remainder.

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24

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

25

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

26

Write the coefficients of the polynomial p(z) = z^{5} – 2z^{2} + 4.

27

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

28

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

29

For what value of k, – 4 is a zero of the polynomial x^{2} – x – (2k + 2)?

30

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

31

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

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32

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

33

For what value of k, is 3 a zero of the polynomial 2x^{2} + x + k?

34

For what value of k, is – 3 a zero of the polynomial x^{2} + 11x + k?

35

For what value of k, is – 2 a zero of the polynomial 3x^{2} + 4x + 2k?

36

If a quadratic polynomial f (x) is factorizable into linear distinct factors, then what is the total number of real and distinct zeros of f(x)?

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37

If a quadratic polynomial f (x) is a square of a linear polynomial, then its two zeroes are coincident. (True/False)

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38

If a quadratic polynomial f(x) is not factorizable into linear factors, then it has no real zero. (True/False)

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39

If f(x) is a polynomial such that f(a)f(b)< 0, then what is the number of zeros lying between a and b?

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40

If graph of quadratic polynomial ax^{2} + bx + c cuts positive direction of y-axis, then what is the sign of c?

41

If the graph of quadratic polynomial ax^{2} + bx + c cuts negative direction of y-axis, then what is the sign of c?

1

If α, β are the zeros of the polynomial f (x) = x^{2} + x + 1, then =

2

If α, β are the zeros of the polynomial p(x) = 4x^{2} + 3x + 7, then =

3

If one zero of the polynomial f (x) = (k^{2} + 4) x^{2} + 13x + 4k is reciprocal of the other, then k =

4

If the sum of the zeros of the polynomial f(x) = 2x^{3} – 3kx^{2} + 4x – 5 is 6, then the value of k is

5

If α and β are the zeros of the polynomial f(x) = x^{2} + px + q, then a polynomial having and is its zeros is

6

If α, β are the zeros of polynomial f(x) = x^{2} – p (x + 1) – c, then (α + 1) (β + 1) =

7

If α, β are the zeros of the polynomial f(x) = x^{2} – p (x + 1) – c such that (α + 1) (β + 1) = 0, then c =

8

If f(x) = ax^{2} + bx + c has no real zeros and a + b + c < 0, then

9

If the diagram in Fig. 2.22 shows the graph of the polynomial f (x) = ax^{2} + bx + c, then

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10

Figure 2.23 shows the graph of the polynomial f(x) = ax^{2} + bx + c for which

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11

If the product of zeros of the polynomial f(x) = ax^{3} –6x^{2} + 11x – 6 is 4, then a =

12

If zeros of the polynomial f (x) = x^{3} – 3px^{2} + qx – r are in A.P., then

13

If the product of two zeros of the polynomial f (x) = 2xZ^{3} + 6x^{2} – 4x + 9 is 3, then its third zero is

14

If the polynomial f(x) = ax^{3} + bx – c is divisible by the polynomial g(x) = x^{2} + bx + c, then ab =

15

In Q. No. 14, c =

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16

If one root of the polynomial f (x) = 5x^{2} + 13x + k is reciprocal of the other, then the value of k is

17

If α, β, γ are the zeros of the polynomial f(x) = ax^{3} + bx^{2} + cx + d, then =

18

If α, β, γ are the zeros of the polynomial f(x) = ax^{3} + bx^{2} + cx + d, then α^{2} + β^{2} + γ^{2} =

19

If α, β, γ are the zeros of the polynomial f(x) = x^{3} – px^{2} + qx – r, then =

20

If α, β are the zeros of the polynomial f(x) = ax^{2} + bx + c, then =

21

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

22

If two zeros of x^{3} + x^{2} – 5x – 5 are √5 and –√5 , then its third zero is

23

The product of the zeros of x^{3} + 4x^{2} + x – 6 is

24

What should be added to the polynomial x^{2} – 5x + 4, so that 3 is the zero of the resulting polynomial?

25

What should be subtracted to the polynomial x^{2} – 16x + 30, so that 15 is the zero of the resulting polynomial?

26

A quadratic polynomial, the sum of whose zeroes is 0 and one zero is 3, is

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27

If two zeroes of the polynomial x^{3} + x^{2} – 9x – 9 are 3 and – 3, then its third zero is

28

If √5 and – √5 are two zeroes of the polynomial x^{3} + 3x^{2} – 5x – 15, then its third zero is

29

If x + 2 is a factor of x^{2} + ax + 2b and a + b = 4, then

30

The polynomial which when divided by – x^{2} + x – 1 gives a quotient x – 2 and remainder 3, is