Polynomials

Polynomials Important Questions

Here are some essential Class 10 Mathematics questions for Chapter 2, Polynomials. These carefully curated questions aim to assist students in their preparation for the CBSE Class 10 Mathematics Examination 2023-24. Practicing a diverse range of question types will not only help students clarify their doubts but also enable them to prepare more effectively for the exams. By solving these questions, students can build confidence in their problem-solving skills and enhance their overall performance in the Polynomials chapter.

Table of Content

In Chapter 2 of Class 10 Mathematics, Polynomials you will learn about polynomials, their types, and the concept of degree. Additionally, we will study the zeros of a polynomial and the relationship between zeros and coefficients of quadratic polynomials.

A polynomial is a mathematical expression made up of variables (usually represented by the letters x, y, etc.), coefficients (real numbers), and non-negative integer exponents. The general form of a polynomial is given by:

P(x) = a_{n}x_{n} + a_{n-1}x_{n-1} +a_{n-2}x_{n-2 }+ ………………. + a_{1}x + a_{0}

Where a_{n}, a_{n-1}, a_{n-2}, ……………………, a_{1}, a_{0} are called coefficients of x_{n}, x_{n-1}, x_{n-2}, ….., x and constant term respectively and it should belong to a real number.

(b) -10

(c) -7

(d) -2

**Ans. **(b)

**Explanation: **

Given polynomial** **x^{2} + 3x + k

Also, given that 2 is one of the zeros of the polynomial

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

∵ x = 2 is a zero of p(x)

∴ p(2) =0

⇒ (2)^{2} + 3(2) + k = 0

⇒ 4 + 6 + k = 0

⇒ 10 + k = 0

⇒ k = -10

f(x) = x

(b) -3 -2

(c) 3,2

(d) 3, -2

**Ans. **(d)

**Explanation: **

f(x) = x^{2} + x - 6

= x^{2} - 3x + 2x - 6

= x(x - 3) + 2(x - 3)

= (x + 2)(x - 3)

To find zeros put f(x) =0

∴ x = -2, 3

**Ans. **\( x= \sqrt{{\Large\frac{2}{3}}},\sqrt{{\Large\frac{2}{3}}}\)

**Explanation: **

\(Given, 3x^2- 2\sqrt{6}x + 2 = 0\)

\(⇒ 3x^2- \sqrt{6}x - \sqrt{6}x + 2 = 0\)

\(⇒ \sqrt{3}x(\sqrt{3}x - \sqrt{2} ) - \sqrt{2}(\sqrt{3}x - \sqrt{2} ) = 0\)

\(⇒ (\sqrt{3}x - \sqrt{2} )(\sqrt{3}x - \sqrt{2} ) = 0\)

\(⇒ x = \sqrt{\Large{\frac{2}{3}}},\sqrt{\Large{\frac{2}{3}}}\)

**Ans. **p = 4

**Explanation: **

Given,

px(x - 3) + 9 = 0

or px^{2} - 3px + 9 = 0

Let ⍺, β be the zeroes of the polynomial.

\(\text{Then, α + β = 3 and αβ =} {\Large\frac{9}{p}}\)

Also, α = β

∴ 2α = 3

\(\text{or \space \space α = β} = {\Large\frac{3}{2}}\\[3.5 bp]

∴ {\Large\frac{9}{4}} = {\Large\frac{9}{p}}\)

⇒ p = 4

Hence, the zeroes of the polynomial px(x - 3) + 9 = 0 will be equal when p = 4.

**Ans. **Navdeep is right

**Explanation: **

Yes

On comparing the given equation with ax^{2} + bx + c,

We get

Here a = 2, b = -4, c = 5

\(\text{Sum of zeroes = }{\Large \frac{-b}{a}}={\Large\frac{-(-4)}{2}}={\Large\frac{4}{2}}= 2\)

∴ Navdeep is right.

The chapter on Polynomials helps students develop algebraic skills and prepares them for more advanced topics in higher classes. It also finds practical applications in various fields such as science, engineering, and economics. Understanding polynomials is crucial for building a strong foundation in algebra and problem-solving skills.

**Ans:** Monomial: A polynomial with only one term, such as 5x^{3}, 2y, etc.

- Binomial: A polynomial with two terms, such as 3x + 2, 2y
^{2}- 5, etc. - Trinomial: A polynomial with three terms, such as 4x
^{2}+ 3x + 1, 2y^{3}- 6y^{2}+ 7y, etc. - Multinomial: A polynomial with more than three terms, such as 5x
^{4}+ 2x^{3}- 3x^{2}+ 7x - 1, etc

**Ans:** The degree of a polynomial is the highest power of the variable 'x' (or any other variable) in the expression. To find the degree, identify the term with the highest power of the variable.

**Ans:** Factor Theorem: If P(a) = 0, then (x - a) is a factor of the polynomial P(x).

The Factor Theorem helps in finding factors and roots of polynomials. If we know one root (zero) of a polynomial, we can easily find the corresponding factor. Additionally, it simplifies the process of polynomial factorization.

**Ans:** The degree of the zero polynomial is undefined since it does not have any terms.

**Ans:** The maximum number of roots a polynomial of degree 'n' can have is 'n'.