Question

The given polynomial is \[2-{{y}^{2}}-{{y}^{3}}+2{{y}^{7}}\] . Find the degree of given polynomial
\[\begin{align}
  & a)2 \\
 & b)7 \\
 & c)0 \\
 & d)3 \\
\end{align}\]

Answer 1

Hint: The degree of polynomial is the highest power of variable in any polynomial. Hence we will find the highest power in the polynomial and that will be the degree of the given polynomial.

Complete step by step answer:
Now first let us understand what polynomials are. Now polynomials are nothing but algebraic expressions consisting of variables and coefficient. Now the variables of a polynomial can have only non-negative integral powers.
Let us check some examples of polynomials
${{x}^{2}}+2x+1.$ , ${{x}^{3}}+3x$ , 5 , $2x+3$ , x + 1. $x+{{y}^{2}}+3z$ , $\dfrac{1x}{2}+5y$
Note that
${{x}^{\dfrac{1}{2}}},{{x}^{-3}}+{{x}^{2}}+5x,\dfrac{1}{{{x}^{2}}}+x+3$ are not polynomials .since the power is either not an integer or negative.
Now note that in polynomials the unknown alphabet is called a variable and the numbers multiplied to it are called coefficient.
Consider the example $2{{x}^{2}}+3x+4$ here the variable is x and 2 is said to be coefficient of ${{x}^{2}}$ similarly 3 is said to be the coefficient of x. 4 is called the constant.
Now each polynomial has a degree.
The degree of polynomial is nothing but the highest power of the variable that is available.
Now for example ${{x}^{5}}+4{{x}^{3}}+2$ is a polynomial and the degree of this polynomial is 5, since 5 is the highest power in the polynomial.
Now let us consider the given equation \[2-{{y}^{2}}-{{y}^{3}}+2{{y}^{7}}\]
Now the given polynomial is a polynomial in y.
Now the exponents of y in the polynomial are ${{y}^{2}},{{y}^{3}},{{y}^{7}}$ .
Now we know that 7 > 3 > 2.
Hence we have the greatest power of y in our polynomial is 7.
Hence the degree of polynomial is 7.
Option b is the correct option.

Note:
Note that while checking the degree check the power for each term. If the polynomial is not in general form that is $a{{x}^{n}}+b{{x}^{n-1}}+....c$ then first write it in general form.
For example the polynomial $\left( x+2 \right)\left( x+3 \right)$ is not in its general form since the terms are in multiplication hence open the bracket
$\left( x+2 \right)\left( x+3 \right)={{x}^{2}}+2x+3x+6={{x}^{2}}+5x+6$
Now we can say that the polynomial is of degree 2.