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Searching for NCERT Class 9 Maths Chapter 2 Exercise 2.1 solutions? You’re in the right place! This section provides clear and step-by-step solutions for all the questions in Exercise 2.1 of Chapter 2 – Polynomials. This exercise introduces students to the fundamental concepts of polynomials, including their definitions, types (like monomials, binomials and trinomials) and the degree of a polynomial. It helps learners identify and classify polynomials based on the number of terms and powers of variables. These solutions are designed to build a strong foundation in algebraic expressions, making it easier to understand more complex topics later. Ideal for exam preparation and conceptual clarity, this guide simplifies polynomial basics in a structured and student-friendly way—boosting confidence and encouraging a deeper understanding of algebra.
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Exercise 2.1
1. Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer
(i) \( 4 x^{2}-3 x+7\)
Answer
Polynomial in one variable means that the polynomial contain
onlyone variable. For example:\( \mathrm{P}(\mathrm{x})=\mathrm{x}+1, \mathrm{p}(\mathrm{w})=\mathrm{w}^{2}+1000 \), these are all polynomials in one variable.
\(\mathrm{P}(\mathrm{x})=4 x^{2}-3 x+7\)
There is only one variable x with whole number power so this is a polynomial in one variable
(ii) \( y^{2}+\sqrt{2}\)
Answer
\( \mathrm{p}(\mathrm{y})=y^{2}+\sqrt{2} \)There is only one variable \( \mathrm{y} \) with whole number power so this is a polynomial in onevariable
(iii) \( \sqrt[3]{t}+\sqrt[t]{2} \)
Answer
\( \sqrt[3]{t}+\sqrt[t]{2} \) \(\mathrm{p}(\mathrm{t})=\sqrt[3]{t}+\sqrt[t]{2}\)
There is only variable \( t \) but in \( \sqrt[3]{t} \) power of t is \( \frac{1}{2} \) which is not a whole number so \( \sqrt[3]{t}+\sqrt[t]{2} \) is not a polynomial.
(iv) \(\mathrm{y}+\frac{2}{y} \)
Answer
\( \mathrm{p}(\mathrm{y})=\mathrm{y}+\frac{2}{y} \)There is only one variable y but in \( \frac{2}{y} \) power of y is \(-1\) which is not a whole number so \( \mathrm{y}+\frac{2}{y} \) is not a polymial.
(v) \(x^{10}+y^{3}+t^{50} \)
Answer
\( \mathrm{p}(\mathrm{x},
\mathrm{y},\mathrm{t})=x^{10}+y^{3}+t^{50} \)There are three variable \( \mathrm{x}, \mathrm{y} \) and \(\mathrm{t}\) and their powers are whole number so this is a polynomial in three variable and not in one variable.
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2.
(i) Write the coefficients of \( x^{2} \) in each of the
following: \( 2+x^{2}+x \)
Answer
Coefficient of a variable in an equation is the number that
ismultiplied with that variable. Therefore, for finding out the
coefficient of \( x^{2} \) in theequations given below. Watch for
the numbers multiplied with \( x^{2} \) and include thepositive or
negative sign.Since coefficient is the multiplicative part to a variable in a single term, then
Coefficients of \( x^{2}=1 \)
(ii) Write the coefficients of \( x^{2} \) in each of the
following: \( 2-x^{2}+x^{3}\)
Answer
Coefficient of a variable in an equation is the number that
ismultiplied with that variable. Therefore, for finding out the
coefficient of \( x^{2} \) in theequations given below. Watch for
the numbers multiplied with \( x^{2} \) and include thepositive or
negative sign.Since coefficient is the multiplicative part to a variable in a single term, then
Coefficients of \( x^{2}=-1 \)
(iii) Write the coefficients of \( x^{2} \) in each of the
following: \( \frac{\pi}{2}x^{2}+x \)
Answer
Coefficient of a variable in an equation is the number that
ismultiplied with that variable. Therefore, for finding out the
coefficient of \( x^{2} \) in theequations given below. Watch for
the numbers multiplied with \( x^{2} \) and include thepositive or
negative sign.Since coefficient is the multiplicative part to a variable in a single term, then
Coefficients of \( x^{2}=\frac{\pi}{2} \)
(iv) Write the coefficients of \( x^{2} \) in each of the
following: \( \sqrt{2 x}-1\)
Answer
Coefficient of a variable in an equation is the number that
ismultiplied with that variable. Therefore, for finding out the
coefficient of \( x^{2} \) in theequations given below. Watch for
the numbers multiplied with \( x^{2} \) and include thepositive or
negative sign.Since coefficient is the multiplicative part to a variable in a single term, then
Coefficients of \( x^{2}=0 \)
3. Give one exampleeach of a binomial of degree 35, and of a
monomial of degree 100.
Answer
Monomial is a polynomial with only one term. For Example: \(
p(x)=x\) Binomial is a polynomial with two terms. for example: \( \mathrm{p}(\mathrm{x})=3\mathrm{x}+2 x^{212121} \). So it does not matter what the power is just the number of terms should be 2.
And the degree of a polynomial is the highest power of the polynomial
Example of binomial with degree 35 is: \( 3 x^{35}+7 x^{35}+4 \)
Example: of monomial degree is: \(4 x^{100}\), \(y^{100}\)
4.
(i) Write the degree of each of the following polynomials: \( 5
x^{3}+4x^{2}+7 x \)
Answer
\( 5 x^{3} \) has the highest power in the given polynomial which
power is 3. Therefore, the degree of the polynomial is 3
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(ii) Write the degree of each of the following polynomials: \(
4-y^{2} \)
Answer
\( \mathrm{-y^{2}} \) has the highest power in the given
polynomial which power is 2. Therefore, the degree of the
polynomial is 2
(iii) Write the degree of each of the following polynomials: \(
5\mathrm{t}-\sqrt{7} \)
Answer
\( 5\mathrm{t} \) has the highest power in the given polynomial
which power is 1. Therefore, the degree of the polynomial is 1
(iv) Write the degree of each of the following polynomials: 3
Answer
There is no variable in the given polynomial. Therefore, the
degree of the polynomial is 0.
5. Classify the following as linear, quadratic and cubic polynomials:
(i) \( x^{2}+\mathrm{x}\)
Answer
Quadratic Polynomial, as its degree or highest power of variable
is 2.
(ii) \(\mathrm{x}-x^{3} \)
Answer
Cubic Polynomial, as its degree or highest power of variable is 3.
(iii) \( y+y^{2}+4 \)
Answer
Quadratic Polynomial, as its degree or highest power of variable
is 2.
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(iv) \( 1+\mathrm{x}\)
Answer
Linear Polynomial, as its degree or highest power of variable is
1.
(v) 3 t
Answer
Linear Polynomial, as its degree or highest power of variable is
1.
(vi) \( r^{2} \)
Answer
Quadratic Polynomial, as its degree or highest power of variable
is 2.
(vii) \( 7 x^{3} \)
Answer
Cubic Polynomial, as its degree or highest power of variable is 3.
