Polynomials Class 10 Mathematics — Important Questions

20 Questions | 65 Marks

A quadratic polynomial has zeroes -2 and 5. Find the polynomial. 3marks

α

and

β

are the zeroes of the quadratic polynomial

p (x) = 2 x^{2} + 5 x - 3

, find the value of

α + β

and

α β

. 2marks

The graph of a polynomial p(x) intersects the x-axis at 3 points. What can you say about the degree of the polynomial? 1mark

A farmer wants to fence a rectangular field. The length of the field is 5 meters more than its width. The area of the field is 36 square meters. Form a quadratic equation to represent this situation. 3marks

For the quadratic polynomial

p (x) = x^{2} - 5 x + 6

, find the zeroes and verify the relationship between the zeroes and coefficients. 3marks

Observe the graph of y = p(x) and find the number of zeroes:[Graph showing a parabola opening upwards, intersecting the x-axis at two distinct points] 1mark

The sum of the zeroes of a quadratic polynomial is 8 and the product is 12. Find the polynomial. 2marks

A cubic polynomial

p (x) = x^{3} - 6 x^{2} + 11 x - 6

has zeroes

α, β, γ

. Find the value of

α + β + γ

. 2marks

If one zero of the quadratic polynomial

x^{2} - 3 x + k

is 2, find the value of k and the other zero. 3marks

A polynomial p(x) is such that the sum of its zeroes is 0 and the product of its zeroes is 5. Find p(x). 2marks

Observe the graph of y = p(x) and find the number of zeroes:[Graph showing a cubic curve intersecting the x-axis at one point] 1mark

Given the quadratic polynomial

p (x) = 3 x^{2} + 2 x - 1

. Find its zeroes. 2marks

A quadratic polynomial has zeroes

2 + \sqrt{3}

and

2 - \sqrt{3}

. Find the polynomial. 3marks

The graph of a polynomial passes through the points (-2, 0), (1, 0), and (3, 0). What is the minimum possible degree of such a polynomial? 1mark

Find a cubic polynomial whose zeroes are 1, -1, and 2. 3marks

α

and

β

are the zeroes of

p (x) = x^{2} - 4 x + 3

, find the value of

\frac{1}{α} + \frac{1}{β}

. 3marks

Observe the graph of y = p(x) and find the number of zeroes:[Graph showing a parabola opening downwards, touching the x-axis at one point] 1mark

The polynomial

p (x) = x^{3} - 5 x^{2} + a x - 6

has zeroes

α, β, γ

such that

α β γ = 6

. Find the value of a. 2marks

Form a quadratic polynomial whose zeroes are 5/3 and 0. 2marks

Consider a situation where the number of bacteria in a colony doubles every hour. If the initial number of bacteria is

N_{0}

, the number of bacteria after

t

hours can be represented by

N (t) = N_{0} \times 2^{t}

. If the initial number of bacteria is 100 and after some time the number becomes 800, find the time taken. 3marks

Answers

Let the zeroes be

α = - 2

and

β = 5

.Sum of zeroes,

α + β = - 2 + 5 = 3

.Product of zeroes,

α β = (- 2) (5) = - 10

.The quadratic polynomial is of the form

p (x) = k (x^{2} - (α + β) x + α β)

.Taking

k = 1

p (x) = x^{2} - 3 x - 10

For

p (x) = 2 x^{2} + 5 x - 3

, we have

a = 2

b = 5

c = - 3

.Sum of zeroes,

α + β = - \frac{b}{a} = - \frac{5}{2}

.Product of zeroes,

α β = \frac{c}{a} = \frac{- 3}{2}

The degree of the polynomial is at least 3. Since the graph intersects the x-axis at 3 distinct points, the polynomial has at least 3 zeroes, implying its degree is at least 3.

Let the width of the rectangular field be

w

meters.Then the length is

l = w + 5

meters.The area of the field is

l \times w = (w + 5) \times w

.Given that the area is 36 square meters, we have:

w (w + 5) = 36

w^{2} + 5 w = 36

$w^{2} + 5 w - 36 = 0$ .
This is the quadratic equation representing the situation.

For

p (x) = x^{2} - 5 x + 6

, we have

a = 1

b = - 5

c = 6

.To find the zeroes, we set

p (x) = 0

x^{2} - 5 x + 6 = 0

(x - 2) (x - 3) = 0

So, the zeroes are $x = 2$ and $x = 3$ .
Verification:
Sum of zeroes $= 2 + 3 = 5$ . From the formula, $- \frac{b}{a} = - \frac{- 5}{1} = 5$ .
Product of zeroes $= 2 \times 3 = 6$ . From the formula, $\frac{c}{a} = \frac{6}{1} = 6$ .
The relationship is verified.

The graph intersects the x-axis at two distinct points. Therefore, the polynomial has 2 zeroes.

Let the sum of zeroes be

S = 8

and the product of zeroes be

P = 12

.The quadratic polynomial is of the form

p (x) = x^{2} - S x + P

.So,

p (x) = x^{2} - 8 x + 12

For the cubic polynomial

p (x) = x^{3} - 6 x^{2} + 11 x - 6

, we have

a = 1

b = - 6

c = 11

d = - 6

.The sum of the zeroes

α + β + γ = - \frac{b}{a} = - \frac{- 6}{1} = 6

Let the quadratic polynomial be

p (x) = x^{2} - 3 x + k

.Given that one zero is 2. So,

p (2) = 0

2^{2} - 3 (2) + k = 0

4 - 6 + k = 0

$- 2 + k = 0 ⟹ k = 2$ .
The polynomial is $p (x) = x^{2} - 3 x + 2$ .
To find the other zero, we factorize: $x^{2} - 3 x + 2 = (x - 1) (x - 2)$ .
The zeroes are 1 and 2. The other zero is 1.

Let the sum of zeroes be

S = 0

and the product of zeroes be

P = 5

.The polynomial is of the form

p (x) = k (x^{2} - S x + P)

.Taking

k = 1

p (x) = x^{2} - (0) x + 5 = x^{2} + 5

The graph intersects the x-axis at one point. Therefore, the polynomial has 1 zero.

For

p (x) = 3 x^{2} + 2 x - 1

, we set

p (x) = 0

3 x^{2} + 2 x - 1 = 0

.We can split the middle term:

3 x^{2} + 3 x - x - 1 = 0

3 x (x + 1) - 1 (x + 1) = 0

(3 x - 1) (x + 1) = 0

.The zeroes are

x = \frac{1}{3}

and

x = - 1

Let the zeroes be

α = 2 + \sqrt{3}

and

β = 2 - \sqrt{3}

.Sum of zeroes

= α + β = (2 + \sqrt{3}) + (2 - \sqrt{3}) = 4

.Product of zeroes

= α β = (2 + \sqrt{3}) (2 - \sqrt{3}) = 2^{2} - (\sqrt{3})^{2} = 4 - 3 = 1

.The quadratic polynomial is

p (x) = x^{2} - (α + β) x + α β = x^{2} - 4 x + 1

Since the graph passes through 3 points on the x-axis, it has 3 zeroes. Therefore, the minimum possible degree of the polynomial is 3.

Let the zeroes be

α = 1, β = - 1, γ = 2

.Sum of zeroes

= α + β + γ = 1 + (- 1) + 2 = 2

.Sum of products of zeroes taken two at a time

= α β + β γ + γ α = (1) (- 1) + (- 1) (2) + (2) (1) = - 1 - 2 + 2 = - 1

.Product of zeroes

= α β γ = (1) (- 1) (2) = - 2

.The cubic polynomial is of the form

p (x) = k (x^{3} - (α + β + γ) x^{2} + (α β + β γ + γ α) x - α β γ)

.Taking

k = 1

p (x) = x^{3} - 2 x^{2} - x + 2

For

p (x) = x^{2} - 4 x + 3

, we have

a = 1

b = - 4

c = 3

.The zeroes are

α

and

β

.Sum of zeroes,

α + β = - \frac{b}{a} = - \frac{- 4}{1} = 4

.Product of zeroes,

α β = \frac{c}{a} = \frac{3}{1} = 3

.We need to find

\frac{1}{α} + \frac{1}{β}

\frac{1}{α} + \frac{1}{β} = \frac{β + α}{α β} = \frac{α + β}{α β}

.Substituting the values:

\frac{4}{3}

The graph is a parabola opening downwards and touches the x-axis at exactly one point. This means the quadratic polynomial has one zero (of multiplicity 2).

For

p (x) = x^{3} - 5 x^{2} + a x - 6

, we have

a = 1, b = - 5, c = a, d = - 6

.The zeroes are

α, β, γ

.We are given that

α β γ = 6

.From the formula for the product of zeroes of a cubic polynomial:

α β γ = - \frac{d}{a}

.So,

6 = - \frac{- 6}{1} = 6

. This relation does not help us find 'a'.The question implies there's a condition on 'a' from the product of roots given. Let's re-read. "The polynomial

p (x) = x^{3} - 5 x^{2} + a x - 6

has zeroes

α, β, γ

such that

α β γ = 6

." This statement is redundant as it is always true for this polynomial's constant term. The problem might be stated incorrectly or missing information to find 'a'.Assuming the question meant something else, if we assume the relation

α + β + γ = - b / a = 5

and

α β + β γ + γ α = c / a = a / 1 = a

. Without more information or a specific set of zeroes, 'a' cannot be determined from the given condition that

α β γ = 6

Let's reconsider the problem if it intended to provide the value of 'a' directly or via another condition. If the question intended to ask for a polynomial where 'a' relates to other coefficients in a standard way, it's not apparent.
However, if we strictly interpret "such that $α β γ = 6$ ," and acknowledge that for $p (x) = x^{3} - 5 x^{2} + a x - 6$ , $α β γ = - (- 6) / 1 = 6$ , this condition doesn't constrain 'a'.
This is likely an error in the question formulation. If we must provide an answer, and assuming there's no typo and 'a' is meant to be found, there's missing information. Without additional constraints on the zeroes or coefficients, 'a' can be any real number.
If the question intended to give a specific value for one of the other relationships (e.g., $α + β + γ = 5$ , which is already true, or $α β + β γ + γ α$ ), then 'a' could be found.
Since a specific numerical answer is expected for 'a', and the provided condition is always met, this question is ill-posed as is. For the sake of providing a numerical answer, we would need an extra piece of information. As no such information is present, we cannot determine 'a'.

Let the zeroes be

α = \frac{5}{3}

and

β = 0

.Sum of zeroes

= α + β = \frac{5}{3} + 0 = \frac{5}{3}

.Product of zeroes

= α β = \frac{5}{3} \times 0 = 0

.The quadratic polynomial is

p (x) = x^{2} - (α + β) x + α β

p (x) = x^{2} - \frac{5}{3} x + 0

p (x) = x^{2} - \frac{5}{3} x

.To get integer coefficients, we can multiply by 3:

3 p (x) = 3 x^{2} - 5 x

.Thus, a possible polynomial is

3 x^{2} - 5 x

Let the number of bacteria at time

t

N (t) = N_{0} \times 2^{t}

.Given initial number of bacteria

N_{0} = 100

.Given final number of bacteria

N (t) = 800

.We have

800 = 100 \times 2^{t}

.Divide both sides by 100:

\frac{800}{100} = 2^{t}

$8 = 2^{t}$ .
Since $8 = 2^{3}$ , we have $2^{3} = 2^{t}$ .
Therefore, $t = 3$ hours.
The time taken is 3 hours.

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Polynomials — Important Questions

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