Class 9 Maths Chapter 2 Polynomials NCERT Solutions | CBSE Complete Guide with Step-by-Step Solutions

Class 9 Maths Chapter 2

Polynomials – NCERT Solutions with Step-by-Step Explanation

Polynomials are an important part of algebra. In Class 9 Maths Chapter 2 Polynomials, students learn how algebraic expressions are formed using variables, constants and powers.

This chapter explains the idea of polynomials, their degrees, types of polynomials and how to find the value of a polynomial for a given value of the variable. Students also learn about the zeroes of a polynomial and how these zeroes relate to the graph of the polynomial.

Understanding polynomials is important because this concept is used in many later topics like factorisation, quadratic equations and calculus.

What is a Polynomial?

A polynomial in one variable $x$ is an expression of the form

$$a_n x^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0$$

where

• $a_n, a_{n-1}, ..., a_0$ are constants

• $n$ is a non-negative integer

Example

$$2x^3 + 3x^2 - 5x + 7$$

Important Terms

Constant

A number without a variable.

Example

$$5,\; -3,\; 10$$

Variable

A symbol representing a number.

Example

$$x,\; y,\; a$$

Degree of a Polynomial

The highest power of the variable in a polynomial is called its degree.

Example

$$4x^3 + 2x^2 + 5$$

Degree = $3$

Types of Polynomials

Constant Polynomial

$$5$$

Degree $=0$

Linear Polynomial

$$ax + b$$

Degree $=1$

Example

$$2x + 3$$

Quadratic Polynomial

$$ax^2 + bx + c$$

Degree $=2$

Example

$$3x^2 + 5x + 2$$

Cubic Polynomial

$$ax^3 + bx^2 + cx + d$$

Degree $=3$

Example

$$x^3 + 2x^2 + x + 1$$

Zero of a Polynomial

A real number $k$ is called a zero of a polynomial $p(x)$ if

$$p(k) = 0$$

Example

$$p(x) = x - 3$$

If $x = 3$

$$p(3) = 3 - 3 = 0$$

So 3 is a zero of the polynomial.

Exercise 2.1 Solutions

Question 1

Which of the following expressions are polynomials?

(i)

$$5x^2 - 3x + 7$$

Since powers of $x$ are non-negative integers.

Answer: Polynomial

(ii)

$$4x^{-1} + 2$$

Since power of $x$ is negative.

Answer: Not a polynomial

Question 2

Write the degree of each polynomial.

(i)

$$7x^4 - 3x^2 + 5$$

Highest power of $x$ is 4.

Degree = 4

(ii)

$$3x^3 + 2x + 1$$

Highest power of $x$ is 3.

Degree = 3

Exercise 2.2 Solutions

Question

Find the value of the polynomial

$$p(x) = 2x^2 - 3x + 1$$

when $x = 2$

Solution

Substitute $x = 2$

$$p(2) = 2(2)^2 - 3(2) + 1$$ $$= 2(4) - 6 + 1$$ $$= 8 - 6 + 1$$ $$= 3$$

Question

Find the value of

$$p(x) = x^3 - 2x^2 + x$$

when $x = -1$

Solution

$$p(-1) = (-1)^3 - 2(-1)^2 + (-1)$$ $$= -1 - 2(1) - 1$$ $$= -4$$

Exercise 2.3 Solutions

Question

Find the zero of the polynomial

$$p(x) = x - 5$$

Solution

Set

$$p(x) = 0$$ $$x - 5 = 0$$ $$x = 5$$

Zero of polynomial = 5

Question

Find the zero of

$$p(x) = 2x + 3$$

Solution

$$2x + 3 = 0$$ $$2x = -3$$ $$x = -\frac{3}{2}$$

Zero of polynomial

$$-\frac{3}{2}$$

Graph of a Polynomial

The zero of a polynomial is the point where the graph intersects the x-axis.

For example

$$p(x) = x - 2$$

The graph cuts the x-axis at $x = 2$.

Key Points to Remember

• A polynomial contains variables with non-negative integer powers.
• The highest exponent of the variable gives the degree.
• Linear polynomial has degree 1.
• Quadratic polynomial has degree 2.
• A number $k$ is a zero if $p(k) = 0$.
• Zeroes of a polynomial correspond to x-intercepts of the graph.

Frequently Asked Questions (FAQs)

1. What is a polynomial?

A polynomial is an algebraic expression made of variables and constants with non-negative integer powers.

2. What is the degree of a polynomial?

The highest power of the variable in the polynomial.

3. What is a linear polynomial?

A polynomial of degree 1.

4. What is a quadratic polynomial?

A polynomial of degree 2.

5. What is a cubic polynomial?

A polynomial of degree 3.

6. What is a constant polynomial?

A polynomial with no variable.

7. What is the zero of a polynomial?

A value of $x$ that makes $p(x)=0$.

8. Can a polynomial have negative powers?

No.

9. Is $x^{-1}$ a polynomial term?

No.

10. Is $5$ a polynomial?

Yes. It is a constant polynomial.

11. What is the degree of $3x^4 + 2x^2$?

Degree = 4.

12. How many zeroes can a polynomial have?

It depends on its degree.

13. What is the graph of a linear polynomial?

A straight line.

14. What is the graph of a quadratic polynomial?

A parabola.

15. Why are polynomials important?

They are widely used in algebra, physics, engineering and higher mathematics.

Conclusion

Class 9 Maths Chapter 2 Polynomials introduces one of the most important ideas in algebra. Once you understand how polynomials are formed, how their degrees are determined and how to find their zeroes, solving algebraic problems becomes much easier.

Practice every question carefully and try solving them yourself before looking at the solution.

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