Exercise 2.1: Polynomials — NCERT Solutions

A polynomial in one variable must have only whole-number exponents of the variable.

(i) $4x^2 - 3x + 7$: exponents are $2, 1, 0$ — all whole numbers. It is a polynomial.

(ii) $y^2 + \sqrt{2}$: exponent of $y$ is $2$; $\sqrt{2}$ is just a constant coefficient. It is a polynomial.

(iii) $3\sqrt{t} + t\sqrt{2} = 3t^{1/2} + \sqrt{2}\,t$: the term $3t^{1/2}$ has exponent $\tfrac12$, which is not a whole number. It is not a polynomial.

(iv) $x + \dfrac{2}{x} = x + 2x^{-1}$: exponent $-1$ is not a whole number. It is not a polynomial.

The coefficient of $x^2$ is the number multiplying $x^2$.

(i) $5 - x^2 + 4x$: the $x^2$ term is $-x^2 = -1\cdot x^2$, so coefficient $= -1$.

(ii) $\sqrt{2}\,x^2 - 3x$: coefficient $= \sqrt{2}$.

(iii) $\dfrac{\pi}{2}x^2 + x$: coefficient $= \dfrac{\pi}{2}$.

(iv) $\sqrt{3}\,x - 5$: there is no $x^2$ term, so coefficient $= 0$.

The degree is the highest power of the variable appearing in the polynomial.

(i) $7x^3 - 4x^2 + x - 9$: highest power is $3$, so degree $= 3$.

(ii) $5 - y^2$: highest power is $2$, so degree $= 2$.

(iii) $3$: a non-zero constant is written as $3x^0$, so degree $= 0$.

(iv) $x^4 - 2x^2 + 6$: highest power is $4$, so degree $= 4$.