# Polynomials

## Class 9 NCERT Maths

### NCERT

1   Which of the following expressions are polynomials in one variable and which are not? State reasons for your answer.$\\$ (i)$4x^2-3x+7$$\\ (ii)y^2+\sqrt{2}$$\\$ (iii)$3\sqrt{t}+t\sqrt{2}$$\\ (iv)y+\dfrac{2}{y}$$\\$ (v)$y+2y^{-1}$

##### Solution :

(i)$4x^2-3x+7$$\\ One variable is involved in given polynomial which is ‘x’ Therefore, it is a polynomial in one variable ‘x’.\\ (ii)y^2+\sqrt{2}$$\\$ One variable is involved in given polynomial which is ‘y’ Therefore, it is a polynomial in one variable ‘y’. $\\$ (iii)$3\sqrt{t}+t\sqrt{2}$$\\ No. It can be observed that the exponent of variable t in term 3\sqrt{t} \ is \ \dfrac{1}{2},which is nota whole number. Therefore, this expression is not a polynomial.\\ (iv)y+\dfrac{2}{y}$$\\$ $=y+2y^{-1}$$\\ The power of variable ‘y’ is -1 which is not a whole number. Therefore, it is not a polynomial in one variable\\ No. It can be observed that the exponent of variable y in term \dfrac{2}{y} is -1 which is not a whole number. Therefore, this expression is not a polynomial.\\ (v)x^{10}+y^3+t^{50}$$\\$ In the given expression there are 3 variables which are ‘x, y, t’ involved.$\\$ Therefore, it is not a polynomial in one variable.

2   Write the coefficients of $x^2$ in each of the following:$\\$ (i)$2+x^2+x$$\\ (ii)2-x^2+x^3$$\\$ (iii)$\dfrac{\pi}{2}x^2+x$$\\ (iv)\sqrt{2x}-1 ##### Solution : (i)2+x^2+x\\ =2+1(x^2)+x$$\\$ The coefficient of $x^2$ is $1$.$\\$ (ii)$2-x^2+x^3 =2-1(x^2)+x$$\\ The coefficient of x^2 is -1.\\ (iii)\dfrac{\pi}{2}x^2+x$$\\$ The coefficient $x^2$ of is $\dfrac{pi}{2}$$\\ (iv)\sqrt{2x}-1 =0x^2+\sqrt{2x}-1$$\\$ The coefficient of $x^2$ is $0$

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

##### Solution :

Binomial of degree $35$ means a polynomial is having$\\$ 1. Two terms$\\$ 2. Highest degree is $35$$\\ Example: x^{35}+x^{34}$$\\$ Monomial of degree $100$ means a polynomial is having $\\$ 1. One term$\\$ 2. Highest degree is $100 $$\\ Example : x^{100} . 4 Write the degree of each of the following polynomials:\\ (i)5x^3+4x^2+7x$$\\$ (ii)$4-y^2$$\\ (iii)5t-\sqrt{7}$$\\$ (iv)$3$

##### Solution :

Degree of a polynomial is the highest power of the variable in the polynomial.$\\$ (i)$5x^3+4x^2+7x$$\\ Highest power of variable ‘x’ is 3. Therefore, the degree of this polynomial is 3$$\\$ (ii)$4-y^2$$\\ Highest power of variable ‘y’ is 2. Therefore, the degree of this polynomial is 2.\\ (iii)5t-\sqrt{7}$$\\$ Highest power of variable ‘t’ is $1$. Therefore, the degree of this polynomial is $1$.$\\$ (iv)$3$$\\ This is a constant polynomial. Degree of a constant polynomial is always 0. 5 Classify the following as linear, quadratic and cubic polynomial:\\ (i)x^2+x$$\\$ (ii)$x-x^3$$\\ (iii)y+y^2+4$$\\$ (iv)$1+x$$\\ (v)3t$$\\$ (vi)$r^2$$\\ (vii)7x^2 7x^3$$\\$

##### Solution :

Linear polynomial - whose variable power is ‘1’$\\$ Quadratic polynomial - whose variable highest power is ‘2’$\\$ Cubic polynomial- whose variable highest power is ‘3’$\\$ (i) $x^ 2 + x$ is a quadratic polynomial as its highest degree is $2$.$\\$ (ii) $x - x^ 3$ is a cubic polynomial as its highest degree is $3.$$\\ (iii) y + y^2 + 4 is a quadratic polynomial as its highest degree is 2.$$\\$ (iv) $1 + x$ is a linear polynomial as its degree is $1.$$\\ (v) 3t is a linear polynomial as its degree is 1.\\ (vi) r^2 is a quadratic polynomial as its degree is 2.$$\\$ (vii) $7x^2 7x^3$ is a cubic polynomial as highest its degree is $3.$