CBSE Class 9 Maths Important Questions - Chapter 2 Polynomials

Multiple Choice Questions (MCQs) from Chapter 2: Polynomials for Class 9 Maths 

1. Which of the following is the degree of the polynomial $5x^4 - 3x^3 + x - 6$?

a) 4
b) 3
c) 1
d) 2

Answer: a) 4

2. What is the factorization of $x^2 + 5x + 6$?

a) (x+3)(x+2)
b)(x+1)(x+6)
c) (x−1)(x+6)
d) (x−3)(x−2)

Answer: a)(x+3)(x+2)

3. Which of the following is not a polynomial?

a) $x^3 + 4x^2 - 2x + 1$
b) $5x^4 - 3x + 2$
c) $x\frac{1}{x} + 3x$

d) $7x^2 - 3x + 4$

Answer: c) $x\frac{1}{x} + 3x$

4. If x+2 is a factor of the polynomial$x^2 + 3x + k$, then the value of k is:

a) -6
b) 4
c) -4
d) 6

Answer: c) -4

5. The zeros of the polynomial $x^2 - 5x + 6$ are:

a) 2 and 3
b) -2 and -3
c) 1 and 6
d) -1 and -6

Answer: a) 2 and 3

6. What is the value of p(3) for the polynomial $p(x) = 4x^2 - 3x + 7$?

a) 30
b) 24
c) 27
d) 40

Answer: b) 24

7. Which of the following is the factorization of $x^2 - 16$?

a) (x−4)(x+4)
b) (x−2)(x+8)
c) (x−8)(x+2)
d) (x−1)(x+16)

Answer: a) (x−4)(x+4)

8. If x=−3 is a root of the polynomial $x^2 + 6x + 9$, then the factor is:

a) x+3

b) x−3
c) x+6
d) x−9

Answer: a) x+3

9. What is the remainder when $x^3 - 4x^2 + 2$ is divided by x−2?

a) -8
b) -12
c) 0
d) 12

Answer: b) -12

10. Which of the following is a polynomial of degree 3?

a) $x^4 - 2x^3 + 3x^2 - x + 2$
b) $x^3 + 2x - 5$
c) $5x^2 + 4x - 3$
d) $x^2 + 2x + 1$

Answer: b) $x^3 + 2x - 5$

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

a) $5x^2 - 3x + 1$
b) $x^2 + 1$
c) $\frac{1}{x}$
d) $x^3 + 2x^2 + 7$

Answer: c) $\frac{1}{x}$

12. What is the factor of the polynomial $x^2 + 7x + 12$?

a) (x+3)(x+4)
b) (x+2)(x+6)
c) (x−3)(x−4)
d) (x+1)(x+12)

Answer: a) (x+3)(x+4)

13. Which of the following polynomials is a binomial?

a) $x^2 + 3x + 2$
b) 4x−7
c) $3x^3 + 2x - 1$
d) $x^2 + 2x + 1$

Answer: b) 4x−7

14. What is the constant term in the polynomial $2x^3 + 4x^2 - 3x + 5$?

a) 4
b) -3
c) 5
d) 2

Answer: c) 5

15. If $p(x) = 3x^2 - 5x + 2$, what is the value of p(2)?

a) 4
b) 5
c) 6
d) 3

Answer: a) 4

Important Questions from Chapter 2: Polynomials For Class 9 Maths

1. What is the difference between a polynomial and a rational expression?

Answer: A polynomial is an algebraic expression involving sums and products of variables raised to non-negative integer powers with constant coefficients, whereas a rational expression is a ratio of two polynomials. A rational expression can involve division by a polynomial, whereas a polynomial cannot have division by a variable.

2. Verify if x - 2 is a factor of $x^2 - 4x + 4$.

Answer: To verify if x−2 is a factor, divide $x^2 - 4x + 4$ by x−2. If the remainder is zero, then x−2 is a factor.

• Factorize $x^2 - 4x + 4$ as (x−2).

• Since x−2 is a factor, the verification is successful.

3. Find the value of k if x+3 is a factor of $x^2 + kx - 18$.

Answer: Use the Factor Theorem. If x+3 is a factor, then x=−3 should satisfy the equation.

Substitute x=−3 into $x^2 + kx - 18$:

$(-3)^2 + k(-3) - 18 = 0(−3)$

9−3k−18=0 

⇒−3k=9

⇒k=−3

4. What are the coefficients of the polynomial $4x^3 + 2x^2 - 5x + 6$?

Answer: The coefficients of the polynomial $x^3 + 2x^2 - 5x + 64$ are:

• Coefficient of $x^3$ = 4

• Coefficient of $x^2$ = 2

• Coefficient of x= -5

• Constant term = 6

5. Find the value of a if the polynomial $2x^2 + ax + 5$ has a factor of x+1.

Answer: By the Factor Theorem, if x+1 is a factor, then x=−1 should satisfy the equation$2x^2 + ax + 5 = 0$.

Substitute x=−1 into $2x^2 + ax + 5$: 

$2(-1)^2 + a(-1) + 5 = 0$

$2 - a + 5 = 0$

−a+7=0

⇒a=7 

6. Find the factorization of $x^2 - 10x + 21$

Answer: To factorize $x^2 - 10x + 21$, find two numbers that multiply to 21 and add to −10. The numbers are −3 and −7.

Therefore, the factorization is:$x^2 - 10x + 21 = (x - 3)(x - 7)$

7. If $p(x) = 3x^2 - 5x + 2$, find p(2).

Answer: Substitute x=2 into $p(x) = 3x^2 - 5x + 2$: 

$p(2) = 3(2)^2 - 5(2) + 2 = 3(4) - 10 + 2 = 12 - 10 + 2 = 4

So, p(2)=4

8. Find the remainder when $x^3 - 4x^2 + 2x - 8$ is divided by x−2.

Answer: Using the Remainder Theorem, substitute x=2 into the polynomial $x^3 - 4x^2 + 2x - 8$:

$2^3 - 4(2^2) + 2(2) - 8 = 8 - 16 + 4 - 8 = -12$ So, the remainder is −12.

9. Explain the concept of the degree of a polynomial and find the degree of the polynomial $5x^4 - 3x^3 + x - 6$.

Answer: The degree of a polynomial is the highest power of the variable in the polynomial.

For $x^4 - 3x^3 + x - 6$, the highest power of x is 4.

Therefore, the degree of the polynomial is 4.

10. Is $x^3 + 2x^2 - 5x + 4$ a polynomial? Justify your answer.

Answer: Yes, $x^3 + 2x^2 - 5x + 4$ is a polynomial because it is an expression in which the exponents of xxx are non-negative integers (3, 2, 1, and 0), and it contains no division by a variable or negative exponents.

11. Solve for the zeros of $x^2 + 6x + 9$.

Answer: The polynomial $x^2 + 6x + 9$ can be factored as: 

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

Therefore, the only zero of the polynomial is x=−3.

12. Find the value of the polynomial $3x^2 + 4x - 5$ at x=−2.

Answer: Substitute x=−2 into the polynomial $x^2 + 4x - 5$: 

$3(-2)^2 + 4(-2) - 5 = 3(4) - 8 - 5 = 12 - 8 - 5 = -1$ 

Therefore, the value is −1.

13. How do you check whether x−1 is a factor of $x^2 - 3x + 2$?

Answer: To check whether x−1 is a factor of $x^2 - 3x + 2$, substitute x=1 into the polynomial. 

$1^2 - 3(1) + 2 = 1 - 3 + 2 = 0$ Since the result is zero, x−1 is indeed a factor of the polynomial.

14. Factorize $x^2 - 16$.

Answer: $x^2 - 16$ is a difference of squares, and can be factorized as: $x^2 - 16 = (x - 4)(x + 4)$

15. What is the constant term in the polynomial $5x^2 - 4x + 7$?

Answer: The constant term in the polynomial $5x^2 - 4x + 7$ is 7.

Class 9 Maths Chapter 2 Important Questions

Students are presented with an extensive view of the algebraic concepts and theories in class 9 Maths Ch 2 important questions. To explore different concepts of the chapter and practice a variety of problems, students must have their hands on important polynomials for class 9. Now, let's discuss some of the details about the chapter:

Polynomials

Polynomials can be quoted as an algebraic expression formed using indeterminates or variables and constants or coefficients. This algebraic expression allows such to perform addition, subtraction, multiplication, positive integer exponentiation of variables. The word polynomial is framed from 'poly' meaning 'many' and 'nominal' meaning 'term,' depicting many terms, which means a polynomial contains many terms but not infinite terms.

A polynomial expression comprises variables like x, y,z, coefficients like 1,2, and exponents like 2 in x². The polynomial function is generally depicted by P(x), where x is the variable. For instance,

P(x) = x² + 7x + 15, here x is the variable and 15 is the constant.

Types of Polynomials

Polynomials are categorised into three groups depending upon the number of terms it comprises of. Here are the types of polynomials.

Monomial

A monomial is a type of polynomial in algebra consisting of a single non-zero term. A polynomial expression consists of one or more terms. Therefore, every term of a polynomial expression is a monomial. Every numeric value such as 6, 12, 151 is a monomial by itself, whereas the variables can x, y, a can also be included in the list of monomials in algebra. Example of a monomial expression – 7x².

Rules for monomial algebraic expressions:

If a monomial is multiplied by a constant, the output will also be a monomial.

If a monomial is multiplied by a monomial, the result will also be a monomial. For instance, if a monomial three is multiplied by 3, the result 8 is also a monomial.

Binomial

A binomial is a type of polynomial expression comprising of two non-zero terms. Let's see some examples to make it clear,

7x² + 8y is a binomial expression with two variables.

10x⁴ + 9y is also a binomial expression with two variables.

Trinomial

A trinomial is a type of polynomial expression comprising of three non-zero terms. Let's see some examples to make it clear,

5x²+8x+9 is a trinomial expression with one variables x.

a + b+ c is a trinomial expression with three variables.

7x – 6y + 9z is a trinomial expression with three variables.

Students can explore different questions polynomial types through the class 9 Maths chapter 2 important questions.

Polynomial Theorems

Some of the vital theorems of polynomials are as follows:

Remainder Theorem

The polynomial remainder theorem, also quoted as the little Bezout's theorem, implies that if a polynomial P(x) is divided by any linear polynomial depicted by (x – a), the remainder of the operation will be a constant given by P(a), i.e., r = P(a).

Factor Theorem

The factor theorem implies that if P(x) is a polynomial of degree n > 1, and 'a' is a real number, this portrays that:

If P(x) = 0, then (x – a) is the factor of P(x),

If (x – a) is the factor of P(x), P(x) = 0.

Bezout's Theorem

Bezout's Theorem states that if P(x) = 0, then P(x) gets divided by (x – a), with 'r' as the remainder.

Intermediate Value Theorem

The intermediate value theorem states that when a polynomial function transforms from a negative to a positive value, it must cross the x-axis. In other words, the theorem highlights the properties of continuity of a function.

Fundamental Theorem of Algebra

The fundamental theorem of algebra states that each non-constant single variable that consists of a complex coefficient possess a minimum of one complex root.

Polynomial Equations

A polynomial equation is an algebraic equation comprising of variables with positive integer exponents and constants. A polynomial expression may contain many exponents, and the highest exponent value is termed as the degree of the equation. Let's take an example to make it clear,

ax⁴ + bx² + x + c, is a polynomial expression with degree = 4.

Algebraic identities of polynomials

• Identity 1 : (x + z )2 = x² + 2xz + z²

• Identity 2 : (y – z) 2 = y² – 2yz + z²

• Identity 3 : y² – z² = (y + z) (y – z)

• Identity 4 : (x + y) (x + z) = x² + (y + z)x + yz

Important Questions of Polynomials for Class 9

To present the students an insight into the algebraic world, we have highlighted here some of the important questions class 9 Maths chapter 2, after a proper analysis of sample question papers:

• What is a polynomial? Explain with example.

• What are the types of polynomial expressions?

• Explain the Remainder Theorem with an example.

• Prove the Factor theorem of polynomials.

• Illustrate Bezout's Theorem, and mention it's importance.

• What do you mean by the degree of the polynomial? Explain with examples.

• How can we add or subtract polynomials?

• Explain the standard form of polynomials.

• What do you mean by roots of equations? And how to find them.

• Find the roots of polynomial equation, f(x) = x⁴ + 5x² + 7x + 19.

Important Formulas Covered In Chapter 2 - Polynomials of Class 9 Maths

Benefits of Class 9 Maths Chapter 2 Polynomials Important Questions

The students preparing for the boards in the upcoming year must prepare a strong core foundation for developing an in-depth logic and understanding of algebra. Therefore they can blend the benefits by practising the class 9th Maths chapter 2 important questions. Here we have listed some of the fruitfulness of class 9 polynomials important questions:

• Students can develop deep learning of the topics by exploring different types of questions presented in the important polynomials for class 9.

• Vedantu, with an efficient team of top-notch educators, has carefully designed the questions after proper research and analysis of the past year's question papers and sample test papers.

• The important questions of ch 2 Maths class 9 are carefully designed under the CBSE board's rules’ strict guidance.

• To perform well in mathematics, academic success is practice; the students must efficiently practice the polynomials class 9 important questions.

• To prevent any issues or mistakes in the important questions for class 9 maths polynomials, expert teachers have reviewed and analysed the papers.

Conclusion

Mathematics is the foundation for logic and reasoning. As a result, in order to grasp the topic's various subjects, students must work with insufficient fundamental Mathematics comprehension and study important polynomial questions for class 9. Students must have a good comprehension of the crucial questions for class 9 mathematics chapter 2 in order to begin a career in science and technology.

Important Study Materials for Class 9 Maths Chapter 2 Polynomials

CBSE Class 9 Maths Chapter-wise Important Questions

CBSE Class 9 Maths Chapter-wise Important Questions and Answers include topics from all chapters. They help students prepare well by focusing on important areas, making revision easier.

Other Important Related Links for CBSE Class 9 Maths