Chapter 2 Polynomials Exercise Ex. 2.1
Question 1(i)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
f(x) = x2 – 2x – 8 Solution 1(i)
x2 – 2x – 8 = x2 – 4x + 2x – 8 = x(x – 4) + 2(x – 4) = (x – 4)(x + 2) The zeroes of the quadratic equation are 4 and -2. Let ∝ = 4 and β = -2 Consider f(x) = x2 – 2x – 8 Sum of the zeroes =
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
g(s) = 4s2 – 4s + 1 Solution 1(ii)
4s2 – 4s + 1 = 4s2 – 2s – 2s + 1 = 2s(2s – 1) – (2s – 1) = (2s – 1)(2s – 1) The zeroes of the quadratic equation are and . Let ∝ = and β = Consider4s2 – 4s + 1 Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(iii)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
h (t) = t2 – 15 Solution 1(iii)
h (t) = t2 – 15 = (t + √15)(t – √15) The zeroes of the quadratic equation areand . Let ∝ = and β = Considert2 – 15 = t2 – 0t – 15 Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(iv)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
f(s) = 6x2 – 3 – 7x Solution 1(iv)
f(s) = 0 6x2 – 3 – 7x =0 6x2 – 9x + 2x – 3 = 0 3x (2x – 3) + (2x – 3) = 0 (3x + 1) (2x – 3) = 0 The zeroes of a quadratic equation are and . Let ∝ = and β = Consider6x2 – 7x – 3 = 0 Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(v)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
Solution 1(v)
The zeroes of a quadratic equation are and . Let ∝ = and β = Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(vi)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
Solution 1(vi)
The zeroes of a quadratic equation are and . Let ∝ = and β = Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(vii)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
Solution 1(vii)
The zeroes of a quadratic equation are and 1. Let ∝ = and β = 1 Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(viii)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
Solution 1(viii)
The zeroes of a quadratic equation are a and. Let ∝ = a and β = Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(ix)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
Solution 1(ix)
The zeroes of a quadratic equation are and . Let ∝ = and β = Consider Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(x)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
Solution 1(x)
The zeroes of a quadratic equation are and . Let ∝ = and β = Consider=0 Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(xi)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
Solution 1(xi)
The zeroes of a quadratic equation are and. Let ∝ = and β = Consider=0 Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 1(xii)
Find the zeros of each of the following quadratic polynomials and verify the relationship between the zeros and their coefficients:
Solution 1(xii)
The zeroes of a quadratic equation are and. Let ∝ = and β = Consider=0 Sum of the zeroes = …(i) Also, ∝ + β = …(ii) Product of the zeroes = …(iii) Also, ∝ β = …(iv) Hence, from (i), (ii), (iii) and (iv),the relationship between the zeroes and their coefficients is verified.Question 2(i)
For 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 factorization.
Solution 2(i)
Question 2(ii)
For 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 factorization.
Solution 2(ii)
Question 2(iii)
For 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 factorization.
Solution 2(iii)
Question 2(iv)
For 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 factorization.
Solution 2(iv)
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Solution 11
Question 12
Solution 12
Question 13
Solution 13
Question 14
Solution 14
Question 15
Solution 15
Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
Solution 21
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Chapter 2 Polynomials Exercise Ex. 2.2
Question 1Verify 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:
Solution 1
On comparing the given polynomial with the polynomial ax3 + bx2 + cx + d, we obtain a = 2, b = 1, c = -5, d = 2
Thus, the relationship between the zeroes and the coefficients is verified.
On comparing the given polynomial with the polynomial ax3 + bx2 + cx + d, we obtain a = 1, b = -4, c = 5, d = -2.
Thus, the relationship between the zeroes and the coefficients is verified.
Concept insight: The zero of a polynomial is that value of the variable which makes the polynomial 0. Remember that there are three relationships between the zeroes of a cubic polynomial and its coefficients which involve the sum of zeroes, product of all zeroes and the product of zeroes taken two at a time.
Question 2
Solution 2
Question 3
Find all zeroes of the polynomial 3x3 + 10x2 – 9x – 4, if one of its zeroes is 1.Solution 3
Let f(x) = 3x3 + 10x2 – 9x – 4
As 1 is one of the zeroes of the polynomial, so (x – 1) becomes the factor of f(x).
Dividing f(x) by (x – 1), we have
Hence, the zeroes are Question 4
If 4 is a zero of the cubic polynomial x3 – 3x2 – 10x + 24, find its other two zeroes.Solution 4
Let f(x) = x3 – 3x2 – 10x + 24
As 4 is one of the zeroes of the polynomial, so (x – 4) becomes the factor of f(x).
Dividing f(x) by (x – 4), we have
Hence, the zeroes are 4, -3 and 2.Question 5
Solution 5
Question 6
Solution 6
Chapter 2 Polynomials Exercise Ex. 2.3
Question 1 (i)
Solution 1 (i)
Question 1 (ii)
Solution 1 (ii)
Question 1 (iii)
Solution 1 (iii)
Question 1 (iv)
Solution 1 (iv)
Question 2
Solution 2
Question 3
Solution 3
Question 4
Solution 4
Question 5
Solution 5
Question 6
Solution 6
Question 7
Solution 7
Question 8
Solution 8
Question 9
Solution 9
Question 10
Solution 10
Question 11
Find all zeros of the polynomial 2x4 – 9x3 + 5x2 + 3x – 1, if two of its zeros are Solution 11
Let f(x) = 2x4 – 9x3 + 5x2 + 3x – 1
As are two of the zeroes of the polynomial, so becomes the factor of f(x).
Dividing f(x) by we have
Hence, the other zeros Question 12
For what value of k, is the polynomial f(x) = 3x4 – 9x3 + x2 + 15x + k completely divisible by 3x2 – 5?Solution 12
Let f(x) = 3x4 – 9x3 + x2 + 15x + k
As f(x) is completely divisible by 3x2 – 5, it becomes one of the factors of f(x).
Dividing f(x) by 3x2 – 5, we have
As (3x2 – 5) is one of the factors, the remainder will be 0.
Therefore, k + 10 = 0
Thus, k = -10.Question 13
Solution 13
Question 14
Solution 14
Chapter 2 Polynomials Exercise 2.61
Question 1
(a) 1
(b) -1
(c) 0
(d) None of theseSolution 1
Question 2
Solution 2
So, the correct option is (d).Question 3
If one zero of the polynomial f(x) = (k2 + 4)x2 + 13x + 4k is reciprocal of the other, then k =
(a) 2
(b) -2
(c) 1
(d) -1Solution 3
Chapter 2 Polynomials Exercise 2.62
Question 4
If the sum of the zeros of the polynomial f(x) = 2x3 – 3kx2 + 4x – 5 is 6, then value of k is
(a) 2
(b) 4
(c) -2
(d) -4Solution 4
So, the correct option is (b).Question 5
(a) x2 + qx + p
(b) x2 – px + q
(c) qx2 + px + 1
(d) px2 + qx + 1Solution 5
Question 6
If α, β are the zeros of polynomial f(x) = x2 – p(x + 1) – c, then (α + 1) (β + 1) =
(a) c – 1
(b) 1 – c
(c) c
(d) 1 + cSolution 6
Question 7
If α, β are the zeros of the polynomial f(x) = x2 – p(x + 1) – c such that (α + 1) (β + 1) = 0 then c =
(a) 1
(b) 0
(c) -1
(d) 2Solution 7
Given that (α + 1) (β + 1) = 0
So, the correct option is (a).Question 8
If f(x) = ax2 + bx + c has no real zeros and a + b + c < 0, then
(a) c = 0
(b) c > 0
(c) c < 0
(d) None of theseSolution 8
We know that, if the quadratic equation ax2 + bx + c = 0 has no real zeros
then
Case 1:
a > 0, the graph of quadratic equation should not intersect x – axis
It must be of the type
Case 2 :
a < 0, the graph will not intersect x – axis and it must be of type
According to the question,
a + b + c < 0
This means,
f(1) = a + b + c
f(1) < 0
Hence, f(0) < 0 [as Case 2 will be applicable]
So, the correct option is (c).Question 9
If the diagram in figure show the graph of the polynomial f(x) = ax2 + bx + c then
(a) a < 0, b < 0 and c > 0
(b) a < 0, b < 0 and c < 0
(c) a < 0, b > 0 and c > 0
(d) a < 0, b > 0 and c < 0Solution 9
Question 10
Figure shows the graph of the polynomial f(x) = ax2 + bx + c for which
(a) a < 0, b > 0 and c > 0
(b) a > 0, b < 0 and c > 0
(c) a < 0, b < 0 and c < 0
(d) a > 0, b > 0 and c < 0Solution 10
Question 11
Solution 11
Question 12
If zeros of the polynomial f(x) = x3 – 3px2 + qx – r are in A.P, then
(a) 2p3 = pq – r
(b) 2p3 = pq + r
(c) p3 = pq – r
(d) None of theseSolution 12
Chapter 2 Polynomials Exercise 2.63
Question 13
Solution 13
So, the correct option is (b).Question 14
Solution 14
Question 15
In Q. No. 14, c =
(a) b
(b) 2b
(c) 2b2
(d) -2bSolution 15
So, the correct option is (c).Question 16
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
Solution 19
Question 20
Solution 20
Question 21
If two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are each equal to zero then the third zero is
Solution 21
Question 22
Solution 22
Question 23
The product of the zeros of x3 + 4x2 + x – 6 is
(a) -4
(b) 4
(c) 6
(d) -6Solution 23
Chapter 2 Polynomials Exercise 2.64
Question 24
What should be added to the polynomial x2 – 5x + 4, so that 3 is the zero of the resulting polynomial ?
(a) 1
(b) 2
(c) 4
(d) 5Solution 24
We know that, if α and β are roots of ax2 + bx + c = 0 then they must satisfy the equation.
According to the question, the equation is
x2 – 5x + 4 = 0
If 3 is the root of equation it must satisfy equation.
x2 – 5x + 4 = 0
but f(3) = 32 – 5(3) + 4 = -2
so, 2 has to be added in the equation.
So, the correct option is (b).Question 25
What should be subtracted to the polynomial x2 – 16x + 30, so that 15 is the zero of resulting polynomial?
(a) 30
(b) 14
(c) 15
(d) 16Solution 25
We know that, if α and β are roots of ax2 + bx + c = 0, then α and β must satisfy the equation.
According to the question, the equation is
x2 – 16x + 30 = 0
If 15 is a root, then it must satisfy the equation x2 – 16x + 30 = 0,
But f(15) = 152 – 16(15) + 30 = 225 – 240 + 30 = 15
and so 15 should be subtracted from the equation.
So, the correct option is (c).Question 26
A quadratic polynomial, the sum of whose zeros is 0 and one zero is 3, is
(a) x2 – 9
(b) x2 + 9
(c) x2 + 3
(d) x2 – 3Solution 26
Question 27
If two zeros of the polynomial x3 + x2 – 9x – 9 are 3 and -3, then its third zero is
(a) -1
(b) 1
(c) -9
(d) 9Solution 27
Question 28
Solution 28
Question 29
If x + 2 is a factor of x2 + ax + 2b and a + b = 4, then
(a) a = 1, b = 3
(b) a = 3, b = 1
(c) a = -1, b = 5
(d) a = 5, b = -1Solution 29
Question 30
The polynomial which when divided by -x2 + x – 1 gives a quotient x – 2 and remainder is 3, is
(a) x3 – 3x2 + 3x – 5
(b) -x3 – 3x2 – 3x – 5
(c) -x3 + 3x2 – 3x + 5
(d) x3 – 3x2 – 3x + 5Solution 30
We know that
Dividend = Divisor × quotient + remainder
Then according to question,
Required polynomial
= (-x2 + x – 1) (x – 2) + 3
= -x3 + 2x2 + x2 -2x – x + 2 + 3
= -x3 + 3x2 – 3x + 5
So, the correct option is (c).Question 31
The number of polynomials having zeroes -2 and 5 is
a. 1
b. 2
c. 3
d. more than 3Solution 31
Correct option: (d)
The polynomials having -2 and 5 as the zeroes can be written in the form
k(x + 2)(x – 5), where k is a constant.
Thus, number of polynomials with roots -2 and 5 are infinitely many, since k can take infinitely many values.Question 32
If one of the zeroes of the quadratic polynomial (k – 1)x2 + kx + 1 is – 3, then the value of k is
a.
b.
c.
d. Solution 32
Question 33
The zeroes of the quadratic polynomial x2 + 99x + 127 are
a. Both positive
b. Both negative
c. both equal
d. One positive and one negativeSolution 33
The zeroes of the quadratic polynomial x2 + 99x + 127 are both negative since all terms are positive.
Hence, correct option is (b).Question 34
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 = -6Solution 34
Question 35
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.
b.
c. 0
d. Solution 35
Question 36
The zeroes of the quadratic polynomial x2 + ax + a, a ≠ 0,
a. cannot both be positive
b. cannot both be negative
c. are always unequal
d. are always equalSolution 36
Question 37
If one of the zeros of the cubic polynomial x3 + ax2 + bx + x is -1, then the product of other two zeroes is
a. b – a + 1
b. b – a – 1
c. a – b + 1
d. a – b – 1Solution 37
Chapter 2 Polynomials Exercise 2.65
Question 38
Given that two of the zeros of the cubic polynomial ax3 + bx2 + cx + d are 0, the third zero is
a.
b.
c.
d. Solution 38
Question 39
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. -5Solution 39
Question 40
If the zeros 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 signSolution 40
It is given that the zeros of the quadratic polynomial ax2 + bx + c, c ≠ 0 are equal.
⇒ Discriminant = 0
⇒ b2 – 4ac = 0
⇒ b2 = 4ac
Now, b2 can never be negative,
Hence, 4ac also can never be negative.
⇒ a and c should have same sign.
Hence, correct option is (c).Question 41
If one of the zeros of a quadratic polynomial of the form x2 + ax + b is the negative of the other, then it
a. has no linear term and 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 positiveSolution 41
Question 42
Which of the following is not the graph of a quadratic polynomial?
Solution 42
The graph of a quadratic polynomial crosses X-axis at atmost two points.
Hence, correct option is (d).
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