Polynomials

Action Plan Notes Questions Mock-Test

Study Action Plan (Sorted by Difficulty)

Zeroes of Polynomials

Accuracy: 53%

Total Attempted499

Total Correct264

Total Incorrect235

Difficulty Level:

Medium

Geometrical Meaning of Zeroes

Accuracy: 65%

Total Attempted932

Total Correct602

Total Incorrect330

Difficulty Level:

Medium

Relationship between Zeroes and Coefficients

Accuracy: 70%

Total Attempted562

Total Correct393

Total Incorrect169

Difficulty Level:

Medium

Maximum Number of Zeroes of a Polynomial

Accuracy: 72%

Total Attempted656

Total Correct472

Total Incorrect184

Difficulty Level:

Medium

Polynomial Factorization

Accuracy: 79%

Total Attempted918

Total Correct725

Total Incorrect193

Difficulty Level:

Low

Types and Standard Forms of Polynomials

Accuracy: 79%

Total Attempted886

Total Correct701

Total Incorrect185

Difficulty Level:

Low

Degree of Polynomials

Accuracy: 87%

Total Attempted374

Total Correct324

Total Incorrect50

Difficulty Level:

Low

Forming a Quadratic Polynomial from Sum and Product of Zeroes

Accuracy: 90%

Total Attempted559

Total Correct501

Total Incorrect58

Difficulty Level:

Low

Learning Objectives

Top

Learning Objectives

Understand the concept of the degree of polynomials and identify the highest power of the variable in polynomial expressions.
Calculate the zeroes of polynomials and determine the values of the variable that make the polynomial equal to zero.
Interpret the geometrical meaning of zeroes by identifying the x-coordinates where the graph of the polynomial intersects the x-axis.
Explore the relationship between zeroes and coefficients of quadratic and cubic polynomials, and derive the formulas connecting them.
Perform polynomial factorization using methods such as splitting the middle term or applying identities to express polynomials as products of linear factors.
Analyze the graphical representation of polynomials, including linear, quadratic, and cubic polynomials, to understand how they intersect the axes and their overall shape.

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Revision Notes & Summary

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Chapter Notes

Degree of Polynomials

Definition: The degree of a polynomial is the highest power of the variable in the polynomial expression.
Examples:
- $4x + 2$ is a polynomial of degree 1.
- $2y^2 - 3y + 4$ is a polynomial of degree 2.
- $5x^3 - 4x^2 + x \sqrt{2}$ is a polynomial of degree 3.
- $7u^6 - 3u^4 + 4u^2 + u - 8$ is a polynomial of degree 6.

Zeroes of Polynomials

Definition: Zeroes of a polynomial are the values of the variable that make the polynomial equal to zero.
Example: For $p(x) = x^2 - 3x - 4$ , the zeroes are $-1$ and $4$ because $p(-1) = 0$ and $p(4) = 0$ .

Geometrical Meaning of Zeroes

Linear Polynomials: The zero of a linear polynomial $ax + b$ is the x-coordinate where the graph intersects the x-axis.
Quadratic Polynomials: The zeroes are the x-coordinates of the points where the parabola intersects the x-axis.
Cubic Polynomials: The zeroes are the x-coordinates where the cubic curve intersects the x-axis.

Relationship between Zeroes and Coefficients

Quadratic Polynomials:
- For $ax^2 + bx + c$ $a x^{2} + b x + c$ , if $\alpha$ $α$ and $\beta$ $β$ are the zeroes, then:
  - Sum of zeroes: $\alpha + \beta = -\frac{b}{a}$
  - Product of zeroes: $\alpha \beta = \frac{c}{a}$
Cubic Polynomials:
- For $ax^3 + bx^2 + cx + d$ $a x^{3} + b x^{2} + c x + d$ , if $\alpha$ $α$ , $\beta$ $β$ , and $\gamma$ $γ$ are the zeroes, then:
  - Sum of zeroes: $\alpha + \beta + \gamma = -\frac{b}{a}$
  - Sum of product of zeroes taken two at a time: $\alpha \beta + \beta \gamma + \gamma \alpha = \frac{c}{a}$
  - Product of zeroes: $\alpha \beta \gamma = -\frac{d}{a}$

Polynomial Factorization

Quadratic Polynomials: Factorization by splitting the middle term.
- Example: $2x^2 - 8x + 6 = (2x - 2)(x - 3)$
Cubic Polynomials: Factorization involves finding zeroes and expressing as products of linear factors.

Graphical Representation of Polynomials

Linear Polynomials: Graph is a straight line intersecting the x-axis at one point.
Quadratic Polynomials: Graph is a parabola that can intersect the x-axis at two points, one point, or not at all.
Cubic Polynomials: Graph can intersect the x-axis at up to three points.

Summary

Polynomials of degrees 1, 2, and 3 are called linear, quadratic, and cubic polynomials, respectively.
A quadratic polynomial can have at most 2 zeroes, and a cubic polynomial can have at most 3 zeroes.
The zeroes of a polynomial are the x-coordinates where its graph intersects the x-axis.

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Practice Test – MCQs, True/False

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Multiple Choice Questions

ax + b

ax^2 + bx + c

ax^3 + bx^2 + cx + d

a + bx

Correct Answer: B

Solution:

The standard form of a quadratic polynomial is

ax^2 + bx + c

, where

a

b

, and

c

are real numbers and

a \neq 0

Chapter Concept:

Types and Standard Forms of Polynomials

-3

-4

Correct Answer: A

Solution:

The sum of the zeroes of a quadratic polynomial

ax^2 + bx + c

is given by

-b/a

. For

x^2 - 3x - 4

, the sum is

-(-3)/1 = 3

Chapter Concept:

Relationship between Zeroes and Coefficients

x^2 + 5x + 6

x^2 - 5x + 6

x^2 + 5x - 6

x^2 - 5x - 6

Correct Answer: B

Solution:

The quadratic polynomial can be formed using the relation

x^2 - (\text{sum of zeroes})x + (\text{product of zeroes})

. Substituting the given values, we get

x^2 - (-5)x + 6 = x^2 + 5x + 6

Chapter Concept:

Forming a Quadratic Polynomial from Sum and Product of Zeroes

x^2 - 7x + 10

x^2 + 7x + 10

x^2 - 7x - 10

x^2 + 7x - 10

Correct Answer: A

Solution:

Using the formula for forming a quadratic polynomial:

x^2 - (\text{sum of zeroes})x + (\text{product of zeroes})

, we substitute the given values to get

x^2 - 7x + 10

Chapter Concept:

Forming a Quadratic Polynomial from Sum and Product of Zeroes

Correct Answer: C

Solution:

A cubic polynomial can have at most 3 zeroes. The polynomial

x^3 - 6x^2 + 11x - 6

can be factored to

(x-1)(x-2)(x-3)

, showing it has 3 zeroes.

Chapter Concept:

Maximum Number of Zeroes of a Polynomial

\alpha + \beta = -\frac{b}{a}

and

\alpha \beta = \frac{c}{a}

\alpha + \beta = \frac{b}{a}

and

\alpha \beta = -\frac{c}{a}

\alpha + \beta = \frac{c}{a}

and

\alpha \beta = -\frac{b}{a}

\alpha + \beta = -\frac{c}{a}

and

\alpha \beta = \frac{b}{a}

Correct Answer: A

Solution:

The relationships are derived from comparing the polynomial

ax^2 + bx + c

with its factorized form

(x - \alpha)(x - \beta)

. Thus,

\alpha + \beta = -\frac{b}{a}

and

\alpha \beta = \frac{c}{a}

Chapter Concept:

Forming a Quadratic Polynomial from Sum and Product of Zeroes

The polynomial has no zeroes.

The polynomial has one zero.

The polynomial has two distinct zeroes.

The polynomial has three zeroes.

Correct Answer: C

Solution:

The polynomial

x^2 - 5x + 6

can be factored as

(x-2)(x-3)

, indicating it has two distinct zeroes at

x = 2

and

x = 3

Chapter Concept:

Maximum Number of Zeroes of a Polynomial

Correct Answer: D

Solution:

A polynomial of degree 4 can have at most 4 zeroes.

Chapter Concept:

Maximum Number of Zeroes of a Polynomial

-4

-8

Correct Answer: A

Solution:

The sum of the zeroes of a quadratic polynomial

ax^2 + bx + c

is given by

-b/a

. For

2x^2 - 8x + 6

, it is

-(-8)/2 = 4

Chapter Concept:

Polynomial Factorization

-1 and 4

1 and -4

2 and 3

-2 and -3

Correct Answer: A

Solution:

The zeroes of the polynomial

x^2 - 3x - 4

are the values of

x

for which

p(x) = 0

. Solving

x^2 - 3x - 4 = 0

gives the zeroes as -1 and 4.

Chapter Concept:

Types and Standard Forms of Polynomials

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True or False

Correct Answer: True

Solution:

The relationship between the zeroes of a quadratic polynomial and its coefficients is given by the formulas: sum of zeroes

\alpha + \beta = -\frac{b}{a}

and product of zeroes

\alpha \beta = \frac{c}{a}

. This is derived from comparing the polynomial with its factorized form.

Chapter Concept :

Forming a Quadratic Polynomial from Sum and Product of Zeroes

Correct Answer: False

Solution:

A cubic polynomial can have at most three zeroes, as it is a polynomial of degree 3, which means it can intersect the x-axis at up to three points.

Chapter Concept :

Geometrical Meaning of Zeroes

Correct Answer: True

Solution:

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.

Chapter Concept :

Zeroes of Polynomials

Correct Answer: False

Solution:

The zeroes of a quadratic polynomial are related to its coefficients. Specifically, the sum of the zeroes is equal to the negative of the coefficient of the linear term divided by the coefficient of the quadratic term, and the product of the zeroes is equal to the constant term divided by the coefficient of the quadratic term.

Chapter Concept :

Polynomial Factorization

Correct Answer: False

Solution:

A polynomial of degree 3 can have at most three zeroes, as the graph of

y = p(x)

intersects the x-axis at most three times.

Chapter Concept :

Relationship between Zeroes and Coefficients

Correct Answer: False

Solution:

The zeroes of a cubic polynomial are related to its coefficients through the relationships: sum of zeroes =

-b/a

, sum of products of zeroes taken two at a time =

c/a

, and product of zeroes =

-d/a

Chapter Concept :

Polynomial Factorization

Correct Answer: True

Solution:

A polynomial of degree 2 is defined as a quadratic polynomial, and it can have at most two zeroes as indicated by its degree.

Chapter Concept :

Polynomial Factorization

Correct Answer: False

Solution:

A quadratic polynomial, which is of degree 2, can have at most two zeroes as it intersects the x-axis at most at two points.

Chapter Concept :

Zeroes of Polynomials

Correct Answer: True

Solution:

Given the sum of zeroes

\alpha + \beta = -3

and the product

\alpha \beta = 2

, the quadratic polynomial is

x^2 - (\alpha + \beta)x + \alpha \beta = x^2 + 3x + 2

Chapter Concept :

Forming a Quadratic Polynomial from Sum and Product of Zeroes

Correct Answer: True

Solution:

A polynomial of degree

n

can have at most

n

zeroes. Therefore, a polynomial of degree 4 can have at most 4 zeroes.

Chapter Concept :

Maximum Number of Zeroes of a Polynomial

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Polynomials

Study Action Plan (Sorted by Difficulty)

Zeroes of Polynomials

Medium

Geometrical Meaning of Zeroes

Medium

Relationship between Zeroes and Coefficients

Medium

Maximum Number of Zeroes of a Polynomial

Medium

Polynomial Factorization

Low

Types and Standard Forms of Polynomials

Low

Degree of Polynomials

Low

Forming a Quadratic Polynomial from Sum and Product of Zeroes

Low

Learning Objectives

Learning Objectives

Practice, Analyze & Improve

Revision Notes & Summary

Chapter Notes

Degree of Polynomials

Zeroes of Polynomials

Geometrical Meaning of Zeroes

Relationship between Zeroes and Coefficients

Polynomial Factorization

Graphical Representation of Polynomials

Summary

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Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.Which of the following is the standard form of a quadratic polynomial?

Correct Answer: B

Q2.For the quadratic polynomial p(x)=x2−3x−4p(x) = x^2 - 3x - 4p(x)=x2−3x−4, what is the sum of its zeroes?

Correct Answer: A

Q3.Given that the sum and product of the zeroes of a quadratic polynomial are -5 and 6 respectively, which of the following is the correct quadratic polynomial?

Correct Answer: B

Q4.A quadratic polynomial has a sum of zeroes equal to 7 and a product of zeroes equal to 10. Which of the following polynomials matches these conditions?

Correct Answer: A

Q5.If a cubic polynomial is given by p(x)=x3−6x2+11x−6p(x) = x^3 - 6x^2 + 11x - 6p(x)=x3−6x2+11x−6, how many zeroes does it have?

Correct Answer: C

Q6.Which of the following is a correct relationship between the zeroes and coefficients of a quadratic polynomial ax2+bx+cax^2 + bx + cax2+bx+c?

Correct Answer: A

Q7.Consider the polynomial p(x)=x2−5x+6p(x) = x^2 - 5x + 6p(x)=x2−5x+6. Which of the following statements is true regarding its zeroes?

Correct Answer: C

Q8.Consider a polynomial p(x)=x4−5x3+6x2−4x+8p(x) = x^4 - 5x^3 + 6x^2 - 4x + 8p(x)=x4−5x3+6x2−4x+8. What is the maximum number of zeroes this polynomial can have?

Correct Answer: D

Q9.For the quadratic polynomial 2x2−8x+62x^2 - 8x + 62x2−8x+6, what is the sum of its zeroes?

Correct Answer: A

Q10.If p(x)=x2−3x−4p(x) = x^2 - 3x - 4p(x)=x2−3x−4, what are the zeroes of this quadratic polynomial?

Correct Answer: A

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True or False

Q1.The sum and product of the zeroes of a quadratic polynomial ax2+bx+cax^2 + bx + cax2+bx+c are given by −ba-\frac{b}{a}−ab​ and ca\frac{c}{a}ac​ respectively.

Correct Answer: True

Q2.A cubic polynomial can have at most two zeroes.

Correct Answer: False

Q3.The zeroes of a polynomial are the x-coordinates of the points where its graph intersects the x-axis.

Correct Answer: True

Q4.The zeroes of a quadratic polynomial are not related to its coefficients.

Correct Answer: False

Q5.A cubic polynomial can have more than three zeroes.

Correct Answer: False

Q6.The zeroes of a cubic polynomial are unrelated to its coefficients.

Correct Answer: False

Q7.A polynomial of degree 2 is always called a quadratic polynomial, and it can have at most two zeroes.

Correct Answer: True

Q8.A quadratic polynomial can have more than two zeroes.

Correct Answer: False

Q9.The quadratic polynomial with zeroes having a sum of −3-3−3 and a product of 222 is x2+3x+2x^2 + 3x + 2x2+3x+2.

Correct Answer: True

Q10.A polynomial of degree 4 can have at most 4 zeroes.

Correct Answer: True

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