Chapter 2 – Polynomials

Class 10 Mathematics · 27 questions · 21 with answers

Solved examples

Ex. 1Multiple choice

If one zero of the quadratic polynomial x2 + 3x + k is 2, then the value of k is

  • (A)10
  • (B)–10
  • (C)5
  • (D)–5
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Solution : Answer (B) POLYNOMIALS 9

Ex. 2Multiple choice

Given that two of the zeroes of the cubic polynomial ax3 + bx2 + cx + d are 0, the third zero is –b b c d

  • (A)
  • (B)
  • (C)
  • (D)– a a a a
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Solution : Answer (A). [Hint: Because if third zero is α, sum of the zeroes –b =α+0+0= ]

Ex. 1Short answerExercise 2.1

Can x – 1 be the remainder on division of a polynomial p (x) by 2x + 3? Justify your answer.

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Solution : No, since degree (x – 1) = 1 = degree (2x + 3).

Ex. 2True / FalseExercise 2.1

Is the following statement True or False? Justify your answer. If the zeroes of a quadratic polynomial ax2 + bx + c are both negative, then a, b and c all have the same sign. b b

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Solution : True, because – = sum of the zeroes < 0, so that > 0. Also the product a a of the zeroes = > 0.

Ex. 1Short answerExercise 2.2

Find the zeroes of the polynomial x2 + x – 2, and verify the relation between the coefficients and the zeroes of the polynomial.

Ex. 1Short answerExercise 2.3

Find a quadratic polynomial, the sum and product of whose zeroes are 2 and – 2 , respectively. Also find its zeroes.

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Solution : A quadratic polynomial, the sum and product of whose zeroes are 3 3 2 and – 2 is x – 2 x – 2 3 1 x2 – 2 x – = [2x2 – 2 2 x – 3] 2 2 = [2x2 + 2 x – 3 2x – 3] = [ 2 x ( 2 x + 1) – 3 ( 2 x + 1)] = [ 2 x + 1] [ 2 x – 3]

Ex. 2Short answerExercise 2.3

If the remainder on division of x3 + 2x2 + kx +3 by x – 3 is 21, find the quotient and the value of k. Hence, find the zeroes of the cubic polynomial x3 + 2x2 + kx – 18.

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Solution : Let p(x) = x3 + 2x2 + kx + 3 Then, p(3) = 33 + 2 × 32 + 3k + 3 = 21 i.e., 3k = –27 i.e., k = –9 Hence, the given polynomial will become x3 + 2x2 – 9x + 3. Now, x – 3) x3 + 2x2 – 9x +3(x2 + 5x +6 x3 – 3x2 5x2 – 9x +3 5x2 – 15x 6x + 3 6x – 18 So, x3 + 2x2 – 9x + 3 = (x2 + 5x + 6) (x – 3) + 21 i.e., x3 + 2x2 – 9x – 18 = (x – 3) (x2 + 5x + 6) = (x – 3) (x + 2) (x + 3)

Questions

Q1Short answerExercise 2.1

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

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(A)

Q4Multiple choiceExercise 2.1

–4 2 –2

  • (A)
  • (B)
  • (C)
  • (D)3 3 3 3 2. A quadratic polynomial, whose zeroes are –3 and 4, is (A) x2 – x + 12 (B) x2 + x + 12 x2 x (C) – –6 (D) 2x2 + 2x –24 2 2 3. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and –3, then (A) a = –7, b = –1 (B) a = 5, b = –1 (C) a = 2, b = – 6 (D) a = 0, b = – 6 4. The number of polynomials having zeroes as –2 and 5 is (A) 1 (B) 2 (C) 3 (D) more than 3
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(D) 3 3 3 3 2. A quadratic polynomial, whose zeroes are –3 and 4, is (A) x2 – x + 12 (B) x2 + x + 12 x2 x (C) – –6 (D) 2x2 + 2x –24 2 2 3. If the zeroes of the quadratic polynomial x2 + (a + 1) x + b are 2 and –3, then (A) a = –7, b = –1 (B) a = 5, b = –1 (C) a = 2, b = – 6 (D) a = 0, b = – 6 4. The number of polynomials having zeroes as –2 and 5 is (A) 1 (B) 2 (C) 3 (D) more than 3

Q5Multiple choiceExercise 2.1

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

  • (A)–
  • (B)
  • (C)0
  • (D)– a a a
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(B)

Q6Multiple choiceExercise 2.1

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

  • (A)b – a + 1
  • (B)b – a – 1
  • (C)a – b + 1
  • (D)a – b –1
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(A) b – a + 1

(D) a – b –1

Q7Multiple choiceExercise 2.1

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

  • (A)both positive
  • (B)both negative
  • (C)one positive and one negative
  • (D)both equal
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(B) both negative

Q8Multiple choiceExercise 2.1

The zeroes of the quadratic polynomial x + kx + k, k ≠ 0,

  • (A)cannot both be positive
  • (B)cannot both be negative
  • (C)are always unequal
  • (D)are always equal
Q9Multiple choiceExercise 2.1

If the zeroes of the quadratic polynomial ax + bx + c, c ≠ 0 are equal, then

  • (A)c and a have opposite signs
  • (B)c and b have opposite signs
  • (C)c and a have the same sign
  • (D)c and b have the same sign
Q10Multiple choiceExercise 2.1

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

  • (A)has no linear term and the constant term is negative.
  • (B)has no linear term and the constant term is positive.
  • (C)can have a linear term but the constant term is negative.
  • (D)can have a linear term but the constant term is positive.
Q11Multiple choiceExercise 2.1

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

  • (A)
  • (B)
  • (C)
  • (D)POLYNOMIALS 11
Q1Multiple choiceExercise 2.2

Answer the following and justify:

  • (i)Can x2 – 1 be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5?
  • (ii)What will the quotient and remainder be on division of ax2 + bx + c by px3 + qx2 + rx + s, p ≠ 0?
  • (iii)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)?
  • (iv)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)? (v) Can the quadratic polynomial x2 + kx + k have equal zeroes for some odd integer k > 1?
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(i) Can x2 – 1 be the quotient on division of x6 + 2x3 + x – 1 by a polynomial in x of degree 5?

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

(iii) 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)?

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

Q2Multiple choiceExercise 2.2

Are the following statements ‘True’ or ‘False’? Justify your answers.

  • (i)If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.
  • (ii)If the graph of a polynomial intersects the x-axis at only one point, it cannot be a quadratic polynomial.
  • (iii)If the graph of a polynomial intersects the x-axis at exactly two points, it need not be a quadratic polynomial.
  • (iv)If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms. (v) 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. (vi) 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. (vii) The only value of k for which the quadratic polynomial kx2 + x + k has equal zeros is
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(i) If the zeroes of a quadratic polynomial ax2 + bx + c are both positive, then a, b and c all have the same sign.

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

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

(iv) If two of the zeroes of a cubic polynomial are zero, then it does not have linear and constant terms. (v) 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. (vi) 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. (vii) The only value of k for which the quadratic polynomial kx2 + x + k has equal zeros is

Q1Short answerExercise 2.2

1 1 Solution : x2 + x – 2= (6x2 + x – 12) = [6x2 + 9x – 8x – 12] 6 6 6

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(i) No (ii) 0, ax2 + bx + c (iii) deg p (x) < deg g (x) (iv) deg g (x) < deg p(x) (v) No

Q1Short answerExercise 2.2

1 = [3x (2x + 3) – 4 (2x + 3)] = (3x – 4) (2x + 3) 6 6

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(i) No (ii) 0, ax2 + bx + c (iii) deg p (x) < deg g (x) (iv) deg g (x) < deg p(x) (v) No

Q4Short answerExercise 2.2

3 Hence, and – are the zeroes of the given polynomial. 3 2 The given polynomial is x2 + x – 2. 4 3 –1 Coefficient of x The sum of zeroes = + – = =– and 3 2 6 Coefficient of x 2 4 –3 Constant term the product of zeroes = × =–2 = 3 2 Coefficient of x 2

Q1Short answerExercise 2.3

4x2 – 3x – 1 2. 3x2 + 4x – 4 POLYNOMIALS 13

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1, –

Q3Short answerExercise 2.3

5t2 + 12t + 7 4. t3 – 2t2 – 15t 7 3

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–1,

Q5Short answerExercise 2.3

2x2 + x + 6. 4x2 + 5 2 x – 3 2 4

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, 4 3 5 2 4 2 –3 2 1 5 2 1

Q7Short answerExercise 2.3

2s2 – (1 + 2 2 )s + 2 8. v2 + 4 3 v – 15 3 11 2

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, 2

Q9Short answerExercise 2.3

y2 + 5y –5 10. 7y2 – y – 2 3 3

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–2 5,

Q1Long answerExercise 2.3

3 Hence, the zeroes are – and .

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1, –

Q2Long answerExercise 2.3

2

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, –2

Q3Long answerExercise 2.3

2 So, the zeroes of x + 2 x + kx –18 are 3, – 2, – 3.

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–1,

Q1Multiple choiceExercise 2.4

For each of 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 4 21 5

  • (i),
  • (ii),
Q3Short answerExercise 2.4

3 8 16 –3 1 (iii) –2 3, – 9 (iv) , – 2 5 2 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. POLYNOMIALS 15 3. Given that 2 is a zero of the cubic polynomial 6x3 + 2 2 x – 10x – 4 2 , find its other two zeroes.

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,

Q4Short answerExercise 2.4

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.

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k = –3 2 3 Zeroes of 2x4 + x3 – 14x2 + 5x + 6 are 1, –3, 2, – Zeroes of x2 + 2x – 3 are 1, –3

Q5Short answerExercise 2.4

Given that x – 5 is a factor of the cubic polynomial x3 – 3 5x 2 + 13x – 3 5 , find all the zeroes of the polynomial.

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5, 5 2, 5 – 2

Q6Short answerExercise 2.4

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)?

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a = –1, b = –2 1 and 2 are the zeroes of q(x) which are not the zeroes of p(x).