Exercise 2.1 Polynomials NCERT Solutions Class 9

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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

Video 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 \(x, y, t.\)

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

Video 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  \(x^{2} \).

Question 3

Give one example each of a binomial of degree \(35,\) and of a monomial of degree \(100.\)

Solution

Video 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:  \(\begin{align}5 x^{100}\end{align}\)

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

Video 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 \(3 = 3{x^\circ}\) )

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

Video 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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