# Ch. - 2 Polynomials

## 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 $$3.$$

The chapter 2 starts with an introduction to polynomials and a recall of algebraic expressions which we learnt in previous grades. Next, the chapter deals with polynomials in one variable followed by an explanation of the zeros of a polynomial. Thereafter, the chapter explains the remainder theorem with the help of examples. This is followed by an explanation of the factorization of polynomials. Finally, the chapter explains the concept of algebraic identities via some examples and a series of exercise questions are presented to broaden your problem-solving skills.

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