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 $3.$