Question

How do you determine the degree of the polynomial?

Answer 1

Hint: In this problem, we have to determine the degree of the polynomial. A polynomial is an expression consisting of variables and coefficients with some mathematical operations that is addition, multiplication, subtraction, and division. A polynomial is of three types Monomial, Binomial and Trinomial. So, every polynomial has a different type of variable, thus having a different degree. A degree of the polynomial is the highest or greatest power of a variable in a polynomial. So, for different types of polynomials, there are different methods for finding the degree of the polynomial.

Complete step by step answer:
A polynomial is an expression consisting of variables and their coefficient with any four mathematical operations that is addition, multiplication, division, and subtraction. They are of three types, monomial, binomial, and trinomial. The monomial polynomial is those polynomials which contain only one term, binomial polynomials consist only two terms and trinomial polynomials consist of three terms.
Some examples of the polynomial are:
$2x{{y}^{3}}$ -- it is a type of monomial polynomial.
$3+7a$ --- it is a binomial polynomial.
$x+5+yz$ ---- it is a trinomial polynomial.
All these polynomials have some variables and those variables have some exponents that help us to identify the degree of the polynomial. Thus, the degree of the polynomial is the highest power of a variable in the polynomial.
There are different ways to find the degree of each type of polynomial. Let us take some examples for the same.
 Example 1: Let the polynomial be a trinomial polynomial with the equation: $13{{x}^{7}}+12{{x}^{6}}+{{x}^{5}}$ .
As we know, the degree of the polynomial is the highest power of a variable.
Here, the degree of an individual polynomial is the highest power among all the powers of the variables. Thus, in this case, powers are $\text{7,6, and 5}$ and among 7 is the highest power. Therefore, 7 is the degree of the polynomial $13{{x}^{7}}+12{{x}^{6}}+{{x}^{5}}$ .

Example 2: Let the binomial polynomial be: $x{{y}^{2}}+2{{x}^{3}}y$ .
In this case, we have two variables in each term, thus we have to identify the power of each term by adding the powers of each variable for each term and then select the highest power among them.
That is, $x{{y}^{2}}={{x}^{1}}{{y}^{2}}=1+2=3$
${{x}^{3}}y={{x}^{3}}{{y}^{1}}=3+1=4$
As we see the powers of each term are 3, and 4. Among them, 4 is the highest power.
Therefore, 4 is the degree of the polynomial $x{{y}^{2}}+2{{x}^{3}}y$ .

Example 3: let the polynomial be trinomial: $xyz+2{{x}^{6}}y{{z}^{2}}+5x{{y}^{3}}$
In this case, we will add all the power of each variable for each term and the highest power among all is the degree of the polynomial.
$xyz={{x}^{1}}{{y}^{1}}{{z}^{1}}=1+1+1=3$
${{x}^{6}}y{{z}^{2}}={{x}^{6}}{{y}^{1}}{{z}^{2}}=6+1+2=9$
$x{{y}^{3}}={{x}^{1}}{{y}^{3}}{{z}^{0}}=1+3+0=4$
Among the powers, we see that 9 is the highest power.
Therefore, 9 is the degree of the polynomial $xyz+2{{x}^{6}}y{{z}^{2}}+5x{{y}^{3}}$

Note: Always remember that degree is the highest power among all the powers of the variables. For monomial polynomials, the degree is the exponent of the variable. Since the monomial polynomial contains one term, it may contain one variable or more than one variable. So, we have to see that regarding how to find the variable.
Example: $x$ -- in this case degree of the polynomial is 1
Example: $xyz$ --- in this case degree is $3=1+1+1$ , for each variable we add the exponent, to get the required degree
Example: $x{{y}^{3}}$ ---- in this case degree is $1+3=4$ , for each variable we add the exponents, to get the answer.