Question

Identify the following as polynomial and hence write their degrees :

$7x-4x^2{y}^3+5x{y}^4-4$

Answer 1

We have the algebraic expression :

$7x-4x^2{y}^3+5x{y}^4-4$

Step 1 Identification of polynomial :

Since a polynomial is an algebraic expression in which powers of variables in each term are whole number.

In term-1 that is $7x$ , power of variable $x$ is $1$ which is a whole number

In term-2 that is $-4x^2{y}^3$ , power of variable $x$ is $2$ and power of variable $y$ is $3$, which are a whole number

In term-3 that is $5x{y}^4$, power of variable $x$ is $1$ and power of variable $y$ is $4$ , which are a whole number

In term-4 that is $-4$ , which is a constant term

so powers of all the variables in each term are whole number

Therefore, $\mathbf{7}\mathit{x}\mathbf{}\mathbf{-}\mathbf{}\mathbf{4}{\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{3}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{5}\mathit{x}{\mathit{y}}^{\mathbf{4}}\mathbf{}\mathbf{-}\mathbf{}\mathbf{4}$ is a polynomial

Step 2 Finding degree of polynomial :

Since, Degree of a polynomial is the greatest sum of powers of the variables in a term.

Terms	$7x$	$-4x^2{y}^3$	$5x{y}^4$	$-4$
Degree	$1$	$2+3=\mathbf{5}$ (Maximum)	$1+4=\mathbf{5}$ (Maximum)	$0$

So maximum degree of each term of the given polynomial is $5$

Therefore, Degree of the polynomial $\mathbf{7}\mathit{x}\mathbf{}\mathbf{-}\mathbf{}\mathbf{4}{\mathit{x}}^{\mathbf{2}}{\mathit{y}}^{\mathbf{3}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{5}\mathit{x}{\mathit{y}}^{\mathbf{4}}\mathbf{}\mathbf{-}\mathbf{}\mathbf{4}$ is $\mathbf{5}$