What is the coefficient of $x^2$ in the polynomial $3x^2 + 5x + 7$?

Correct Option: A. Solution: The coefficient of $x^2$ in the polynomial $3x^2 + 5x + 7$ is 3.

Given the polynomial $r(t) = t^4 - 5t^3 + 6t^2 - 4t + 8$, what is the coefficient of $t^3$?

Correct Option: A. Solution: The term with $t^3$ in the polynomial $r(t)$ is $-5t^3$. Therefore, the coefficient of $t^3$ is -5.

If the side length of a square is increased from $x$ units to $x + 5$ units, what is the increase in its perimeter?

Correct Option: D. Solution: The perimeter of a square with side length $x$ is $4x$. When the side length is increased to $x + 5$, the new perimeter is $4(x + 5) = 4x + 20$. The increase in perimeter is $20$ units.

If $p(x) = x^3 - 23x^2 + 142x - 120$, which of the following is a factor of $p(x)$?

Correct Option: A. Solution: By trial, $p(1) = 0$, so $x - 1$ is a factor of $p(x)$.

The polynomial $r(t) = t^3 - 2t^2 + t - 2$ is divided by $t - 1$. What is the remainder?

Correct Option: C. Solution: By the Remainder Theorem, the remainder of $r(t)$ divided by $t - 1$ is $r(1)$. Calculating $r(1) = 1^3 - 2(1)^2 + 1 - 2 = -1$. Therefore, the remainder is -1.

If the polynomial $q(y) = y^3 - 3y^2 + 3y - 1$ is divided by $y - 1$, what is the remainder?

Correct Option: A. Solution: By the Remainder Theorem, the remainder of the division of $q(y)$ by $y - 1$ is $q(1) = 1^3 - 3(1)^2 + 3(1) - 1 = 0$. Therefore, the remainder is 0.

Polynomials

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CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

Understand the definition and terminology related to polynomials.
Identify and classify polynomials based on their degree (linear, quadratic, cubic).
Apply the Remainder Theorem and Factor Theorem in polynomial factorization.
Utilize algebraic identities for simplifying and factorizing expressions.
Expand expressions using the identity for the square of a sum.
Factor polynomials using known identities and theorems.

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 2: Polynomials

2.1 Introduction

Study of algebraic expressions, operations, and factorization.
Introduction to polynomials and related terminology.
Overview of the Remainder Theorem and Factor Theorem.
Exploration of algebraic identities and their applications.

2.2 Polynomials in One Variable

Definition of a Variable: A symbol that can take any real value (e.g., x, y, z).
Algebraic Expressions: Examples include 2x, 3x, -x, etc.
Constants vs. Variables: Constants remain unchanged, while variables can change.

Types of Polynomials

Linear Polynomials (Degree 1):
- Form: ax + b (where a ≠ 0)
- Examples: 4x + 5, 2y, 3u
Quadratic Polynomials (Degree 2):
- Form: ax² + bx + c (where a ≠ 0)
- Examples: 5 - y², 4y + 5y²
Cubic Polynomials (Degree 3):
- Form: ax³ + bx² + cx + d (where a ≠ 0)
- Examples: 4x³, 2x³ + 1
General Polynomial (Degree n):
- Form: aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ (where aₙ ≠ 0)
- The zero polynomial has an undefined degree.

Polynomials in Multiple Variables

Examples: x² + y² + xyz (three variables), p² + q¹⁰ + r (three variables).

2.5 Algebraic Identities

Identity I: (x + y)² = x² + 2xy + y²
Identity II: Not specified in the excerpts.
Identity III: Not specified in the excerpts.
Identity IV: (x + a)(x + b) = x² + (a + b)x + ab
Identity V: (x + y + z)² = x² + y² + z² + 2xy + 2yz + 2zx

Examples of Using Identities

Example 11:
- Find products using identities:
  - (x + 3)(x + 3) = x² + 6x + 9
  - (x - 3)(x + 5) = x² + 2x - 15
Example 14:
- Expand (3a + 4b + 5c)² using Identity V.
- Result: 9a² + 16b² + 25c² + 24ab + 40bc + 30ac
Example 16:
- Factorize 4x² + y² + z² - 4xy - 2yz + 4xz using Identity V.
- Result: (2x - y + z)²

Important Notes

The degree of a polynomial indicates its highest exponent.
The structure of polynomials varies based on their degree and number of terms.

CBSE Quiz & Practice Test – MCQs, True/False Questions with Solutions

Multiple Choice Questions

-4

5

3

-2

Correct Answer: A

Solution:

The coefficient of

x^2

in the polynomial

p(x) = x^3 - 4x^2 + 5x - 2

-4

-x^3

4x^2

7x

5x

Correct Answer: D

Solution:

The polynomial

-x^3 + 4x^2 + 7x - 2

does not contain the term

5x

. The terms are

-x^3

4x^2

7x

, and

-2

Correct Answer: A

Solution:

The coefficient of

x^2

in the polynomial

3x^2 + 5x + 7

is 3.

0

2x + 3

5

x^2 - x + 1

Correct Answer: C

Solution:

A constant polynomial is a polynomial of degree 0, which means it does not have any variable terms. The expression

5

is a constant polynomial. The zero polynomial

0

is also considered a constant polynomial, but it is a special case.

-5

-4

Correct Answer: A

Solution:

The term with

t^3

in the polynomial

r(t)

-5t^3

. Therefore, the coefficient of

t^3

is -5.

x^3 - 2x + 1

3y^2 + 5y

\sqrt{x} + 3

t^2 + 4

Correct Answer: C

Solution:

The expression

\sqrt{x} + 3

can be written as

x^{1/2} + 3

, where the exponent

1/2

is not a whole number. Therefore, it is not a polynomial. Hence, option_c is correct.

-4

5

3

-2

Correct Answer: A

Solution:

In the polynomial

r(t) = t^3 - 4t^2 + 5t - 2

, the coefficient of

t^2

-4

. Hence, option_a is correct.

-1

-2

Correct Answer: B

Solution:

The coefficient of

x^2

in the polynomial

-x^3 + 4x^2 + 7x - 2

is 4.

20 square units

25 square units

30 square units

35 square units

Correct Answer: B

Solution:

The area of a square is given by side \times side. Therefore, the area is 5 \times 5 = 25 square units.

2x^2

square units

4x^2

square units

x^2 + 4x

square units

x^2

square units

Correct Answer: B

Solution:

The original area is

x^2

square units. Doubling the side length gives a new side length of

2x

, so the new area is

(2x)^2 = 4x^2

square units.

5

units

10

units

15

units

20

units

Correct Answer: D

Solution:

The perimeter of a square with side length

x

4x

. When the side length is increased to

x + 5

, the new perimeter is

4(x + 5) = 4x + 20

. The increase in perimeter is

20

units.

x - 1

x + 2

x - 3

x + 4

Correct Answer: A

Solution:

By trial,

p(1) = 0

, so

x - 1

is a factor of

p(x)

x - 1

x + 1

x - 2

x + 2

Correct Answer: A

Solution:

By substituting

x = 1

into

f(x)

, we find

f(1) = 1^4 - 4(1)^3 + 6(1)^2 - 4(1) + 1 = 0

. Thus,

x - 1

is a factor of

f(x)

-7

14

-8

1

Correct Answer: A

Solution:

The coefficient of

t^2

in the polynomial

p(t) = t^3 - 7t^2 + 14t - 8

-7

x^2

square units

2x

square units

4x

square units

x^3

square units

Correct Answer: A

Solution:

The area of a square is given by the formula side \times side, which is

x \times x = x^2

square units.

-1

Correct Answer: C

Solution:

By the Remainder Theorem, the remainder of

r(t)

divided by

t - 1

r(1)

. Calculating

r(1) = 1^3 - 2(1)^2 + 1 - 2 = -1

. Therefore, the remainder is -1.

x^3 + x^2 + x + 1

x^4 + 3x^3 + 3x^2 + x + 1

x^2 + 2x + 1

x^3 - 1

Correct Answer: A

Solution:

Using the Factor Theorem,

(x+1)

is a factor if the polynomial evaluates to zero at

x = -1

. For

x^3 + x^2 + x + 1

, substituting

x = -1

gives

(-1)^3 + (-1)^2 + (-1) + 1 = 0

10 units

15 units

20 units

25 units

Correct Answer: C

Solution:

The perimeter of a square is given by 4 times the length of one side. So, 4 \times 5 = 20 units.

Splitting the middle term

Using the quadratic formula

Completing the square

All of the above

Correct Answer: D

Solution:

The polynomial

q(y) = y^2 - 5y + 6

can be factorized by splitting the middle term, using the quadratic formula, or completing the square. All methods are applicable for quadratic polynomials.

x + 1

x^2 - x + 7

Correct Answer: A

Solution:

A constant polynomial is a polynomial that does not contain any variables. The expression '7' is a constant polynomial.

-4

-6

Correct Answer: A

Solution:

In the polynomial

x^3 - 4x^2 + 5x - 6

, the coefficient of

x^2

is -4.

-4

-5

Correct Answer: A

Solution:

The coefficient of x^2 in the polynomial x^3 - 4x^2 + 5x - 6 is -4.

6 units

7 units

8 units

9 units

Correct Answer: B

Solution:

The perimeter of a square is 4 times the length of one side. Therefore, side length = 28/4 = 7 units.

-1

Correct Answer: A

Solution:

By the Remainder Theorem, the remainder of the division of

q(y)

y - 1

q(1) = 1^3 - 3(1)^2 + 3(1) - 1 = 0

. Therefore, the remainder is 0.

x - 1

x - 2

x - 3

x - 4

Correct Answer: A

Solution:

To determine the factor, we use the Factor Theorem. We find

p(1) = 1^3 - 6(1)^2 + 11(1) - 6 = 0

. Therefore,

x - 1

is a factor of

p(x)

x^2

square units

4x

square units

2x

square units

x

square units

Correct Answer: A

Solution:

The area of a square is given by the square of its side length. Therefore, if the side length is

x

units, the area is

x^2

square units.

x - 1

x - 2

x - 3

x - 4

Correct Answer: C

Solution:

To find the factors of

p(x)

, we can use the Factor Theorem. If

p(c) = 0

, then

x - c

is a factor of

p(x)

. Evaluating

p(x)

x = 3

, we have

p(3) = 3^3 - 5(3)^2 + 6(3) = 27 - 45 + 18 = 0

. Thus,

x - 3

is a factor.

(x+3)^2

x^2 + 6x + 9

4x + 12

x^2 + 9

Correct Answer: A

Solution:

The new side length of the square is

x + 3

. Therefore, the new area is

(x+3)^2 = x^2 + 6x + 9

. Hence, option_a is correct.

x - 1

x + 2

x - 3

x + 1

Correct Answer: A

Solution:

By the Factor Theorem, if

p(1) = 0

, then

x - 1

is a factor. Calculating

p(1)

gives 0.

3x^2 + 5x + 7

x + \frac{1}{x}

\sqrt{x} + 3

\frac{3}{y} + y^2

Correct Answer: A

Solution:

A polynomial is an algebraic expression in which the exponents of the variable are whole numbers. The expression

3x^2 + 5x + 7

is a polynomial because all the exponents are whole numbers.

x^3 - x^2 + 4x + 7

3y^2 + 5y

x + 1/x

t^2 + 4

Correct Answer: C

Solution:

The expression

x + 1/x

is not a polynomial because it includes

x^{-1}

, which is not a whole number exponent.

x - 1

x + 1

x - 2

x + 2

Correct Answer: A

Solution:

By trial, we find that p(1) = 0, so x - 1 is a factor of the polynomial.

y - 1

y - 2

y - 3

y + 1

Correct Answer: B

Solution:

The polynomial

y^2 - 5y + 6

can be factored as

(y - 2)(y - 3)

. Therefore, both

y - 2

and

y - 3

are factors. Hence, option_b is correct.

(y - 1)(y - 3)

(y + 1)(y - 3)

(y - 2)(y - 1)

(y - 3)(y + 2)

Correct Answer: A

Solution:

The polynomial

q(y) = y^2 - 4y + 3

can be factorized by finding two numbers that multiply to 3 and add to -4. These numbers are -1 and -3, so

q(y) = (y - 1)(y - 3)

5x + 25

10x + 25

25

x^2 + 10x + 25

Correct Answer: A

Solution:

The original area is

x^2

. The new area is

(x + 5)^2 = x^2 + 10x + 25

. The increase in area is

(x^2 + 10x + 25) - x^2 = 10x + 25

Yes, because

q(3) = 0

No, because

q(3) \neq 0

Yes, because

q(3) = 27

No, because

q(3) = -27

Correct Answer: A

Solution:

Using the Factor Theorem, if

t - 3

is a factor, then

q(3)

should be zero. Calculating

q(3)

gives

3^3 - 9(3)^2 + 27(3) - 27 = 27 - 81 + 81 - 27 = 0

. Thus,

t - 3

is a factor.

28 units

21 units

14 units

35 units

Correct Answer: A

Solution:

The perimeter of a square is given by 4 times the side length. So, 4 x 7 = 28 units.

3

units

6

units

9

units

12

units

Correct Answer: D

Solution:

The original perimeter is

4x

. The new perimeter is

4(x + 3) = 4x + 12

. Thus, the increase in perimeter is

12

units.

-1

-2

Correct Answer: A

Solution:

The coefficient of

x^2

in the polynomial

-x^3 + 4x^2 + 7x - 2

is 4.

4x

units

x^2

units

2x

units

x

units

Correct Answer: A

Solution:

The perimeter of a square is the sum of the lengths of its four sides. Therefore, if each side is

x

units, the perimeter is

4x

units.

-5

-4

Correct Answer: A

Solution:

The coefficient of

x

is the number multiplying

x

, which is 9.

(x - 2)(x - 3)

(x + 2)(x - 3)

(x - 1)(x - 6)

(x + 1)(x - 6)

Correct Answer: A

Solution:

The polynomial x^2 - 5x + 6 can be factorized as (x - 2)(x - 3) by finding two numbers that multiply to 6 and add to -5.

2x^2 + 3x + 1

1/x + x^2

√x + 2

x + 1/x

Correct Answer: A

Solution:

A polynomial has terms with non-negative integer exponents. 2x^2 + 3x + 1 fits this definition.

4x + 8

4x + 2

4x + 4

4x + 6

Correct Answer: A

Solution:

The original perimeter of the square is

4x

. If the side length is increased by 2 units, the new side length becomes

x + 2

. Thus, the new perimeter is

4(x + 2) = 4x + 8

2x + 4

square units

2x + 2

square units

4x + 4

square units

x^2 + 4x + 4

square units

Correct Answer: A

Solution:

The original area of the square is

x^2

and the new area is

(x + 2)^2 = x^2 + 4x + 4

. The increase in area is

(x^2 + 4x + 4) - x^2 = 4x + 4

square units.

x^2 + 3x + 5

1/x + 3

√x + 2

3/y + y^2

Correct Answer: A

Solution:

A polynomial in one variable has whole number exponents. x^2 + 3x + 5 fits this definition.

12 square units

16 square units

20 square units

24 square units

Correct Answer: B

Solution:

The area of a square is calculated as the side length squared. Therefore, the area is 4 \times 4 = 16 square units.

x - 1

x + 2

x - 3

x + 1

Correct Answer: A

Solution:

By substituting

x = 1

into

p(x)

, we find that

p(1) = 0

. Therefore,

x - 1

is a factor of

p(x)

x^3 + x^2 + x + 1

x^4 + 3x^3 + 3x^2 + x + 1

x^2 - 5x + 6

x^3 - 23x^2 + 142x - 120

Correct Answer: A

Solution:

By using the Factor Theorem, we substitute

x = -1

into each polynomial. For

x^3 + x^2 + x + 1

(-1)^3 + (-1)^2 + (-1) + 1 = 0

, indicating that

(x+1)

is a factor.

True or False

Correct Answer: False

Solution:

In the polynomial

x^2 - x + 7

, the coefficient of

x

is -1, not 1.

Correct Answer: True

Solution:

The perimeter of a square is the sum of the lengths of its four sides. If each side is

x

units, the perimeter is

4x

units.

Correct Answer: True

Solution:

The perimeter of a square is calculated as 4 times the length of one side. Therefore, for a side of 10 units, the perimeter is 4 x 10 = 40 units.

Correct Answer: True

Solution:

The polynomial

3y^2 + 5y + 7

consists of three terms:

3y^2

5y

, and

7

Correct Answer: True

Solution:

The polynomial

3y^2 + 5y + 7

consists of three terms:

3y^2

5y

, and

7

Correct Answer: True

Solution:

A polynomial is an algebraic expression with whole number exponents.

x^2 + 2x

is a polynomial in the variable

x

because both terms have whole number exponents.

Correct Answer: True

Solution:

In the polynomial

-x^3 + 4x^2 + 7x - 2

, the coefficient of

x^2

is 4.

Correct Answer: True

Solution:

In the polynomial

x^2 - x + 7

, the term

-x

has a coefficient of

-1

Correct Answer: True

Solution:

The polynomial

3y^2 + 5y

consists of two terms:

3y^2

and

5y

Correct Answer: False

Solution:

The expression

\sqrt{x} + 3

can be written as

x^{1/2} + 3

. Since the exponent

1/2

is not a whole number, it is not a polynomial.

Correct Answer: True

Solution:

The perimeter of a square is calculated as the sum of the lengths of its four sides. For a square with each side measuring 3 units, the perimeter is 4 \times 3 = 12 units.

Correct Answer: True

Solution:

By using the factorization process,

x^3 - 23x^2 + 142x - 120

can be factorized into

(x - 1)(x - 10)(x - 12)

Correct Answer: True

Solution:

The polynomial

x^3 - x^2 + 4x + 7

consists of four terms:

x^3

-x^2

4x

, and

7

Correct Answer: False

Solution:

The expression

\sqrt{x} + 3

can be written as

x^{1/2} + 3

. Since the exponent

1/2

is not a whole number, it is not a polynomial.

Correct Answer: True

Solution:

x^2 + 2x

is a polynomial in one variable,

x

, with whole number exponents.

Correct Answer: False

Solution:

The expression

3/y + y^2

can be rewritten as

3y^{-1} + y^2

. The exponent

-1

is not a whole number, so the expression is not a polynomial.

Correct Answer: True

Solution:

The expression

3y^2 + 5y

is a polynomial because it consists of terms with whole number exponents of the variable

y

Correct Answer: True

Solution:

A polynomial is an algebraic expression that consists of terms with non-negative integer exponents. The expression

x^3 - x^2 + 4x + 7

has whole number exponents and is therefore a polynomial in

x

Correct Answer: True

Solution:

A polynomial is an algebraic expression with whole number exponents. The expression

x^2 - x + 7

fits this definition, making it a polynomial in

x

Correct Answer: True

Solution:

The polynomial

x^3 - 23x^2 + 142x - 120

can be factorized by first finding that

x - 1

is a factor and then further factorizing the quadratic

x^2 - 22x + 120

to get

(x - 10)(x - 12)

Polynomials

AI Learning Assistant

CBSE Learning Objectives – Key Concepts & Skills You Must Know

Learning Objectives

CBSE Revision Notes & Quick Summary for Last-Minute Study

Chapter 2: Polynomials

2.1 Introduction

2.2 Polynomials in One Variable

Types of Polynomials

Polynomials in Multiple Variables

2.5 Algebraic Identities

Examples of Using Identities

Important Notes

Multiple Choice Questions

Q1.If p(x)=x3−4x2+5x−2p(x) = x^3 - 4x^2 + 5x - 2p(x)=x3−4x2+5x−2, find the coefficient of x2x^2x2.

Correct Answer: A

Q2.Which term is not part of the polynomial −x3+4x2+7x−2-x^3 + 4x^2 + 7x - 2−x3+4x2+7x−2?

Correct Answer: D

Q3.What is the coefficient of x2x^2x2 in the polynomial 3x2+5x+73x^2 + 5x + 73x2+5x+7?

Correct Answer: A

Q4.Which of the following is a constant polynomial?

Correct Answer: C

Q5.Given the polynomial r(t)=t4−5t3+6t2−4t+8r(t) = t^4 - 5t^3 + 6t^2 - 4t + 8r(t)=t4−5t3+6t2−4t+8, what is the coefficient of t3t^3t3?

Correct Answer: A

Q6.Which of the following expressions is not a polynomial?

Correct Answer: C

Q7.Given the polynomial r(t)=t3−4t2+5t−2r(t) = t^3 - 4t^2 + 5t - 2r(t)=t3−4t2+5t−2, determine the coefficient of t2t^2t2.

Correct Answer: A

Q8.What is the coefficient of x2x^2x2 in the polynomial −x3+4x2+7x−2-x^3 + 4x^2 + 7x - 2−x3+4x2+7x−2?

Correct Answer: B

Q9.What is the area of a square with each side measuring 5 units?

Correct Answer: B

Q10.A square has a side length of xxx units. If the side length is doubled, what will be the new area of the square?

Correct Answer: B

Q11.If the side length of a square is increased from xxx units to x+5x + 5x+5 units, what is the increase in its perimeter?

Correct Answer: D

Q12.If p(x)=x3−23x2+142x−120p(x) = x^3 - 23x^2 + 142x - 120p(x)=x3−23x2+142x−120, which of the following is a factor of p(x)p(x)p(x)?

Correct Answer: A

Q13.Consider a polynomial f(x)=x4−4x3+6x2−4x+1f(x) = x^4 - 4x^3 + 6x^2 - 4x + 1f(x)=x4−4x3+6x2−4x+1. Which of the following is a factor of f(x)f(x)f(x)?

Correct Answer: A

Q14.Given the polynomial p(t)=t3−7t2+14t−8p(t) = t^3 - 7t^2 + 14t - 8p(t)=t3−7t2+14t−8, find the coefficient of t2t^2t2.

Correct Answer: A

Q15.What is the area of a square with side length xxx?

Correct Answer: A

Q16.The polynomial r(t)=t3−2t2+t−2r(t) = t^3 - 2t^2 + t - 2r(t)=t3−2t2+t−2 is divided by t−1t - 1t−1. What is the remainder?

Correct Answer: C

Q17.Determine which of the following polynomials has (x+1)(x+1)(x+1) as a factor.

Correct Answer: A

Q18.What is the perimeter of a square with each side measuring 5 units?

Correct Answer: C

Q19.Given the polynomial q(y)=y2−5y+6q(y) = y^2 - 5y + 6q(y)=y2−5y+6, which method can be used to factorize it?

Correct Answer: D

Q20.Which of the following is a constant polynomial?

Correct Answer: A

Q21.What is the coefficient of x2x^2x2 in the polynomial x3−4x2+5x−6x^3 - 4x^2 + 5x - 6x3−4x2+5x−6?

Correct Answer: A

Q22.In the polynomial x^3 - 4x^2 + 5x - 6, what is the coefficient of x^2?

Correct Answer: A

Q23.If the perimeter of a square is 28 units, what is the length of one side?

Correct Answer: B

Q24.If the polynomial q(y)=y3−3y2+3y−1q(y) = y^3 - 3y^2 + 3y - 1q(y)=y3−3y2+3y−1 is divided by y−1y - 1y−1, what is the remainder?

Correct Answer: A

Q25.Consider a polynomial p(x)=x3−6x2+11x−6p(x) = x^3 - 6x^2 + 11x - 6p(x)=x3−6x2+11x−6. Which of the following is a factor of p(x)p(x)p(x)?

Correct Answer: A

Q26.What is the area of a square with side length xxx units?

Correct Answer: A

Q27.Consider a polynomial p(x)=x3−5x2+6xp(x) = x^3 - 5x^2 + 6xp(x)=x3−5x2+6x. Which of the following is a factor of p(x)p(x)p(x)?

Correct Answer: C

Q28.Consider a square with side length xxx. If each side of the square is increased by 333 units, what will be the new area of the square?

Correct Answer: A

Q29.If p(x)=x3−6x2+11x−6p(x) = x^3 - 6x^2 + 11x - 6p(x)=x3−6x2+11x−6, which of the following is a factor of p(x)p(x)p(x)?

Correct Answer: A

Q30.Which of the following expressions is a polynomial?

Correct Answer: A

Q31.Which of the following expressions is not a polynomial?

Correct Answer: C

Q32.If the polynomial is x^3 - 23x^2 + 142x - 120, what is one of its factors?

Correct Answer: A

Q33.Consider the polynomial q(y)=y2−5y+6q(y) = y^2 - 5y + 6q(y)=y2−5y+6. Which of the following is a factor of q(y)q(y)q(y)?

Correct Answer: B