# Exercise 2.1 Polynomials NCERT Solutions Class 9

## Chapter 2 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) \begin{align}4 x^{2}-3 x+7\end{align}

(ii) \begin{align}y^{2}+\sqrt{2}\end{align}

(iii) \begin{align}3 \sqrt{t}+t \sqrt{2}\end{align}

(iv)\begin{align} y+\frac{2}{y}\end{align}

(v)\begin{align}x^{10}+y^{3}+t^{50}\end{align}

### Solution

Steps :

(i) \begin{align}4 x^{2}-3 x+7 \rightarrow\end{align} Polynomial in one variable $$x.$$

(ii) \begin{align}y^{2}+\sqrt{2} \rightarrow\end{align} Polynomial in one variable $$y.$$

(iii) \begin{align}3 \sqrt{t}+t \sqrt{2} \rightarrow\end{align}  Not a polynomial, since the power of the variable in the first term is \begin{align}\frac{1}{2}\end{align} which is not a whole number.

(iv) \begin{align}y+\frac{2}{y} \rightarrow \end{align} Not a polynomial since the power of the variable in the second term is $$– 1$$ which is not a whole number.

(v) \begin{align}x^{10}+y^{3}+t^{50} \rightarrow \end{align} Not a polynomial in one variable since there are $$3$$ variables $$x, y, t.$$

## Chapter 2 Ex.2.1 Question 2

Write the coefficients of $${x^2}$$  in each of the following:

(i) \begin{align}2 +x^{2}+x \end{align}

(ii) \begin{align}2-x^{2}+x^{3}\end{align}

(iii) \begin{align}\frac{\pi}{2} x^{2}+x \end{align}

(iv)  \begin{align} \sqrt{2} x-1\end{align}

### Solution

Steps:

(i) \begin{align}&2 +x^{2}+x \\ \end{align}

Coefficient of  \begin{align}x^{2}=1 \end{align}

(ii) \begin{align}2 -x^{2}+x^{3} \end{align}

Coefficient of $$x^{2}=-1$$

(iii) \begin{align}\frac{\pi}{2}\; x^{2}+x \end{align}

Coefficient of \begin{align}x^{2}=\frac{\pi}{2}\end{align}

(iv) \begin{align}\sqrt{2}\; x-1 \end{align}

Coefficient of  $$x^{2}=0$$, since there is no term of  $$x^{2}$$.

## Chapter 2 Ex.2.1 Question 3

Give one example each of a binomial of degree $$35,$$ and of a monomial of degree $$100.$$

### Solution

Steps:

(i) A binomial of degree $$35$$

Binomial means polynomial having only $$2$$ terms. Here the highest degree should be $$35.$$

So, the binomial will look like \begin{align}a x^{35}-b x^{c}\end{align}  where  \begin{align}a \neq 0, b \neq 0 \text { and } 0 \leq c<35\end{align}

Example: \begin{align}3 x^{35}-5\end{align}

(ii) A monomial of degree $$100$$

Monomial means polynomial having only $$1$$ term. Here the highest degree should be $$100.$$ So, the monomial will look like  \begin{align}a x^{100} \text { where } a \neq 0\end{align}

Example:  \begin{align}5 x^{100}\end{align}

## Chapter 2 Ex.2.1 Question 4

Write the degree of each of the following polynomials:

(i) \begin{align}5 x^{3}+4 x^{2}+7 x \end{align}

(ii) \begin{align}4-y^{2}\end{align}

(iii) \begin{align}5 t-\sqrt{7} \end{align}

(iv) \begin{align}3\end{align}

### Solution

Reasoning:

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

Steps:

(i) Degree of \begin{align}5 x^{3}+4 x^{2}+7 x\;\rm{is}\;3\end{align} (the highest power of the variable $$x$$)

(ii) Degree of \begin{align}4-y^{2}\;\rm{is}\;2\end{align} (the highest power of the variable $$y$$)

(iii) Degree of \begin{align}5 t-\sqrt{7}\;\rm{is}\;1\end{align} (the highest power of the variable $$t$$)

(iv) Degree of $$3\;\rm{is}\;0$$ (degree of a constant polynomial is $$0$$. Here $$3 = 3{x^\circ}$$ )

## Chapter 2 Ex.2.1 Question 5

Classify the following as linear, quadratic and cubic polynomials:

(i) \begin{align}x^{2}+x\end{align}

(ii) \begin{align}x-x^{3}\end{align}

(iii) \begin{align}y+y^{2}+4\end{align}

(iv) \begin{align}1+x\end{align}

(v) \begin{align}3 t\end{align}

(vi) \begin{align}r^{2}\end{align}

(vii) \begin{align}7 x^{3}\end{align}

### Solution

Reasoning:

A polynomial of degree one is called a linear polynomial.

A polynomial of degree two is called a quadratic polynomial.

A polynomial of degree three is called a cubic polynomial.

Steps:

(i) \begin{align}x^{2}+x \rightarrow\end{align} Quadratic polynomial since the degree is $$2.$$

(ii) \begin{align}x-x^{3} \rightarrow\end{align} Cubic polynomial since the degree is $$3.$$

(iii) \begin{align}y+y^{2}+4 \rightarrow\end{align} Quadratic polynomial since the degree is $$2.$$

(iv)\begin{align}1+x \rightarrow\end{align} Linear polynomial since the degree is $$1.$$

(v) \begin{align}3 t \rightarrow\end{align} Liner polynomial since the degree is $$1.$$

(vi) \begin{align}r^{2} \rightarrow\end{align} Quadratic polynomial since the degree is $$2.$$

(vii) \begin{align}7 x^{3} \rightarrow\end{align}  Cubic polynomial since the degree is $$3.$$

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