Question

How do you identify the degree of each term of the polynomial and the degree of the polynomial $F\left( x \right)=5+2x+3{{x}^{2}}+4{{x}^{3}}$ ?

Answer 1

Hint: A polynomial’s degree is the highest or the greatest power of a variable in a polynomial equation. This is called the degree of the polynomial. Degree of each term is the power of the variable with non-zero coefficient. If we have a constant , that means the variable of that function is not there. In such cases, we represent it as ${{x}^{0}}$ . According to the law of exponents, we all know that anything to the power of zero is equal to $1$. Let us use these and find out the degree.

Complete step by step solution:
First let us find the degree of each term.
Since we have a constant in $F\left( x \right)$ which is $5$ , we have to multiply it with ${{x}^{0}}$ since this is a function of $x$.And according to the law of exponents ${{x}^{0}}=1.$
 We are basically multiplying it with $1$. So it does not change the value of the function in any way.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow F\left( x \right)=5+2x+3{{x}^{2}}+4{{x}^{3}} \\
 & \Rightarrow F\left( x \right)=5{{x}^{0}}+2{{x}^{1}}+3{{x}^{2}}+4{{x}^{3}} \\
\end{align}$
The degree of the constant term is $0$ since the power that the variable $x$ carries is $0$.
The degree of the second term which is $2x$ is $1$ since the power that the variable $x$ carries is $1$.
The degree of the third term which is $3{{x}^{2}}$ is $2$ since the power that the variable $x$ carries is $2$.
The degree of the third term which is $4{{x}^{3}}$ is $3$ since the power that the variable $x$ carries is $3$, which is also the highest or the greatest power among all the other terms.
So the degree of the polynomial would be the degree of the ${{4}^{th}}$term as it has the highest power.
$\therefore $ The degree of the polynomial $F\left( x \right)=5+2x+3{{x}^{2}}+4{{x}^{3}}$ is $3$.

Note: We should not get confused between degree of each term and degree of the polynomial. These are two different things. We should also remember the definitions of all these. It is important to multiply with ${{x}^{0}}$ whenever there is a constant term. This is good practice and this avoids any sort of confusion. Do not get confused with the number of the term with the power. In a haste, we might write the answer as $4$. But $4$ is the number of the term which has the highest power .