Exercise 2.1 (Revised) - Chapter 2 - Polynomials - Ncert Solutions class 9 - Maths
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NCERT Solutions Class 9 Maths Chapter 2 - Polynomials | Comprehensive Guide
Ex 2.1 Question 1.
Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.
(i) $4 x^2-3 x+7$
(ii) $y^2+\sqrt{2}$
(iii) $3 \sqrt{t}+t \sqrt{2}$
(iv) $y+\frac{2}{y}$
(v) $x^{10}+y^3+t^{50}$
Answer.
(i) $4 x^2-3 x+7$
We can observe that in the polynomial $4 x^2-3 x+7$, we have $x$ as the only variable and the powers of $x$ in each term are a whole number.
Therefore, we conclude that $4 x^2-3 x+7$ is a polynomial in one variable.
(ii) $y^2+\sqrt{2}$
We can observe that in the polynomial $y^2+\sqrt{2}$, we have $y$ as the only variable and the powers of $y$ in each term are a whole number.
Therefore, we conclude that $y^2+\sqrt{2}$ is a polynomial in one variable.
(iii) $3 \sqrt{t}+t \sqrt{2}$
We can observe that in the polynomial $3 \sqrt{t}+t \sqrt{2}$, we have $t$ as the only variable and the powers of $t$ in each term are not a whole number.
Therefore, we conclude that $3 \sqrt{t}+t \sqrt{2}$ is not a polynomial in one variable.
(iv) $y+\frac{2}{y}$
We can observe that in the polynomial $y+\frac{2}{y}$, we have $y$ as the only variable and the powers of $y$ in each term are not a whole number.
Therefore, we conclude that $y+\frac{2}{y}$ is not a polynomial in one variable.
(v) $x^{10}+y^3+t^{30}$
We can observe that in the polynomial $x^{10}+y^3+t^{30}$, we have $x, y$ and $t$ as the variables and the powers of $x, y$ and $t$ in each term is a whole number.
Therefore, we conclude that $x^{10}+y^3+t^{30}$ is a polynomial but not a polynomial in one variable.
Ex 2.1 Question 2.
Write the coefficients of $x^2$ in each of the following:
(i) $2+x^2+x$
(ii) $2-x^2+x^3$
(iii) $\frac{\pi}{2} x^2+x$
(iv) $\sqrt{2} x-1$
Answer.
(i) $2+x^2+x$
The coefficient of $x^2$ in the polynomial $2+x^2+x$ is 1 .
(ii) $2-x^2+x^3$
The coefficient of $x^2$ in the polynomial $2-x^2+x^3$ is -1 .
(iii) $\frac{\pi}{2} x^2+x$
The coefficient of $x^2$ in the polynomial $\frac{\pi}{2} x^2+x$ is $\frac{\pi}{2}$.
(iv) $\sqrt{2} x-1$
The coefficient of $x^2$ in the polynomial $\sqrt{2} x-1$ is 0 .
Ex 2.1 Question 3.
Give one example each of a binomial of degree 35 , and of a monomial of degree 100 .
Answer.
The binomial of degree 35 can be $x^{35}+9$.
The binomial of degree 100 can be $t^{100}$.
Ex 2.1 Question 4.
Write the degree of each of the following polynomials:
(i) $p(x)=5 x^3+4 x^2+7 x$
(ii) $p(y)=4-y^2$
(iii) $f(t)=5 t-\sqrt{7}$
(iv) $f(x)={ }_3$
Answer.
(i) $5 x^3+4 x^2+7 x$
We know that the degree of a polynomial is the highest power of the variable in the polynomial.
We can observe that in the polynomial $5 x^3+4 x^2+7 x$, the highest power of the variable $x$ is 3 .
Therefore, we conclude that the degree of the polynomial $5 x^3+4 x^2+7 x$ is 3 .
(ii) $4-y^2$
We know that the degree of a polynomial is the highest power of the variable in the polynomial.
We can observe that in the polynomial ${ }^{4-y^2}$, the highest power of the variable $y$ is 2 .
Therefore, we conclude that the degree of the polynomial ${ }^{4-y^2}$ is 2 .
(iii) $5 t-\sqrt{7}$
We know that the degree of a polynomial is the highest power of the variable in the polynomial.
We observe that in the polynomial $5 t-\sqrt{7}$, the highest power of the variable $t$ is 1 .
Therefore, we conclude that the degree of the polynomial $5 t-\sqrt{7}$ is 1 .
(iv) 3
We know that the degree of a polynomial is the highest power of the variable in the polynomial.
We can observe that in the polynomial 3 , the highest power of the assumed variable $x$ is 0 .
Therefore, we conclude that the degree of the polynomial 3 is 0 .
Ex 2.1 Question 5.
Classify the following as linear, quadratic and cubic polynomials:
(i) $x^2+x$
(ii) $x-x^3$
(iii) $y+y^2+4$
(iv) $1+x$
(v) $3 t$
(vi) $r^2$
(vii) $7 x^3$
Answer.
(i) $x^2+x$
We can observe that the degree of the polynomial $x^2+x$ is 2 .
Therefore, we can conclude that the polynomial $x^2+x$ is a quadratic polynomial.
(ii) $x-x^3$
We can observe that the degree of the polynomial $x-x^3$ is 3 .
Therefore, we can conclude that the polynomial $x-x^3$ is a cubic polynomial.
(iii) $y+y^2+4$
We can observe that the degree of the polynomial $y+y^2+4$ is 2 .
Therefore, the polynomial $y+y^2+4$ is a quadratic polynomial.
(iv) $1+x$
We can observe that the degree of the polynomial $(1+x)$ is 1 .
Therefore, we can conclude that the polynomial $1+x$ is a linear polynomial.
(v) $3 t$
We can observe that the degree of the polynomial $(3 t)$ is 1 .
Therefore, we can conclude that the polynomial $3 t$ is a linear polynomial.
(vi) $r^2$
We can observe that the degree of the polynomial $r^2$ is 2 .
Therefore, we can conclude that the polynomial $r^2$ is a quadratic polynomial.
(vii) $7 x^3$
We can observe that the degree of the polynomial $7 x^3$ is 3 .
$\text { Therefore, we can conclude that the polynomial } 7 x^3 \text { is a cubic polynomial. }$
