17/05/2015
(A) Main Concepts and Results
• Geometrical meaning of zeroes of a polynomial: The zeroes of a polynomial p(x)
are precisely the x-coordinates of the points where the graph of y = p(x) intersects
the x-axis.
• Relation between the zeroes and coefficients of a polynomial: If α and β are the
zeroes of a quadratic polynomial ax2 + bx + c, then α + β
b
–
a
, αβ
c
a
• If α, β and γ are the zeroes of a cubic polynomial ax3 + bx2 + cx + d, then
α+β+γ
– b
a
, α β + β γ + γ α
c
a
and α β γ
–d
a
• The division algorithm states that given any polynomial p(x) and any non-zero
polynomial g( x), there are polynomials q(x) and r(x) such that
p(x) = g(x) q(x) + r(x), where r(x) = 0 or degree r(x) < degree g(x).
(B) Multiple Choice Questions
Choose the correct answer from the given four options:
Sample Question 1: 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
Solution : Answer (B)
POLYNOMIALS
CHAPTER 2
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POLYNOMIALS 9
Sample Question 2: Given that two of the zeroes of the cubic polynomial
ax3 + bx2 + cx + d are 0, the third zero is
(A)
–b
a (B)
b
a (C)
c
a (D) – d
a
Solution : Answer (A). [Hint: Because if third zero is α, sum of the zeroes
= α + 0 + 0 =
–b
a ]
EXERCISE 2.1
Choose the correct answer from the given four options in the following questions:
1. If one of the zeroes of the quadratic polynomial (k–1) x2 + k x + 1 is –3, then the
value of k is
(A)
4
3 (B)
–4
3
(C)
2
3
(D)
–2
3
2. A quadratic polynomial, whose zeroes are –3 and 4, is
(A) x2 – x + 12 (B) x2 + x + 12
(C)
2
– – 6
2 2
x x
(D) 2x2 + 2x –24
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
5. Given that one of the zeroes of the cubic polynomial ax3 + bx2 + cx + d is zero, the
product of the other two zeroes is
(A) –
c
a (B)
c
a (C) 0 (D) –
b
a
6. 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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10 EXEMPLAR PROBLEMS
7. 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
8. The zeroes of the quadratic polynomial x2 + kx + k, k ≠ 0,
(A) cannot both be positive (B) cannot both be negative
(C) are always unequal (D) are always equal
9. If the zeroes of the quadratic polynomial ax2 + 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
10. 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.
11. Which of the following is not the graph of a quadratic polynomial?
(A) (B)
(C) (D)
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POLYNOMIALS 11
(C) Short Answer Questions with Reasoning
Sample Question 1: Can x – 1 be the remainder on division of a polynomial p (x) by
2x + 3? Justify your answer.
Solution : No, since degree (x – 1) = 1 = degree (2x + 3).
Sample Question 2: 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.
Solution : True, because – b
a
= sum of the zeroes < 0, so that
b
a > 0. Also the product
of the zeroes =
c
a > 0.
EXERCISE 2.2
1. 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?
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.
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12 EXEMPLAR PROBLEMS
(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
1
2
(D) Short Answer Questions
Sample Question 1:Find the zeroes of the polynomial x2 +
1
6
x – 2, and verify the
relation between the coefficients and the zeroes of the polynomial.
Solution : x2 +
1
6
x – 2 =
1
6 (6x2 + x – 12) =
1
6 [6x2 + 9x – 8x – 12]
=
1
6 [3x (2x + 3) – 4 (2x + 3)] =
1
6 (3x – 4) (2x + 3)
Hence,
4
3 and
3
–
2 are the zeroes of the given polynomial.
The given polynomial is x2 +
1
6
x – 2.
The sum of zeroes =
4
3 +
3 –1
–
2 6
= – 2
Coefficient of
Coefficient of
x
x and
the product of zeroes =
4 –3 – 2
3 2
= 2
Constant term
Coefficient of x
EXERCISE 2.3
Find the zeroes of the following polynomials by factorisation method and verify the
relations between the zeroes and the coefficients of the polynomials:
1. 4x2 – 3x – 1 2. 3x2 + 4x – 4
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POLYNOMIALS 13
3. 5t2 + 12t + 7 4. t3 – 2t2 – 15t
5. 2x2 +
7
2
x +
3
4 6. 4x2 + 5 2x – 3
7. 2s2 – (1 + 2 2)s + 2 8. v2 + 4 3v – 15
9. y2 +
3
5
2
y – 5 10. 7y2 –
11
3
y –
2
3
(E) Long Answer Questions
Sample Question 1: Find a quadratic polynomial, the sum and product of whose
zeroes are 2 and
– 3
2 , respectively. Also find its zeroes.
Solution : A quadratic polynomial, the sum and product of whose zeroes are
2 and
3
–
2 is x2 – 2 x
3
–
2
x2 – 2 x
3
–
2 =
1
2 [2x2 – 2 2x – 3]
=
1
2 [2x2 + 2 x – 3 2x – 3]
=
1
2 [ 2 x ( 2 x + 1) – 3 ( 2 x + 1)]
=
1
2 [ 2 x + 1] [ 2 x – 3]
Hence, the zeroes are
1
–
2 and
3
2 .
Sample Question 2: 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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14 EXEMPLAR PROBLEMS
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
21
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)
So, the zeroes of x3+2x2+kx–18are 3, – 2, – 3.
EXERCISE 2.4
1. 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.
(i)
–8
3 ,
4
3 (ii)
21
8 ,
5
16
(iii) –2 3, –9 (iv)
–3
2 5,
1
–
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.
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POLYNOMIALS 15
3. Given that 2 is a zero of the cubic polynomial 6x3 + 2 x2 – 10x – 4 2 , find
its other two zeroes.
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.
5. Given that x – 5 is a factor of the cubic polynomial x3 – 3 5x2 + 13x – 3 5 ,
find all the zeroes of the polynomial.
6. 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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