Question

What is the degree of the following polynomial expression:-
\[{{u}^{\dfrac{-1}{2}}}\ \ +\ \ 3u\ \ +\ \ 2\] ?
(A). 1
(B). 0
(C). \[\dfrac{-1}{2}\]
(D). Not defined

Answer 1

HINT:- Before solving this question, we must know about Polynomials and degrees.
POLYNOMIALS: A polynomial is defined as an expression that contains two or more algebraic terms.
For example: \[6x{{y}^{2}}\ \ -\ \ {{x}^{3}}\ \ +\ \ 6{{y}^{3}}\ \ -\ \ 5xyz\] is an example of a polynomial expression having 4 algebraic terms.

Complete step-by-step solution -
DEGREE OF POLYNOMIALS: A polynomial’s degree is the highest or the greatest degree of a variable in a polynomial equation. The degree indicates the highest exponential power in the polynomial.
To find the degree of a polynomial, we need to see the exponent of the variable of an expression. The highest exponent is the degree of the whole expression, or we can say the degree of the polynomial. If there is more than one variable in a term of the expression, then we shall add all the exponents of one term to find the degree of that particular term. And then we can find the degree of the whole expression or polynomial.
Now, if the exponent of any variable is a negative number; or the variable is a numerator of a fraction, then that expression is not considered a polynomial, and thus, its degree cannot be found.
Let us now solve this question.
We have to find the degree of the expression \[{{u}^{\dfrac{-1}{2}}}\ \ +\ \ 3u\ \ +\ \ 2\] .
We can see that the very first term of this expression is \[{{u}^{\dfrac{-1}{2}}}\]. We can see that the exponent of the variable ‘u’ is a negative number. So, this cannot be a polynomial, and therefore, its degree cannot be found.
Therefore, the correct option for this question is (d) Not defined.

NOTE:- Let us learn to find the degree of a polynomial with the help of an example.
For example: \[6{{x}^{3}}{{y}^{3}}\ \ -\ \ {{x}^{3}}{{y}^{2}}\ \ +\ \ 6{{y}^{3}}\ \ -\ \ 5{{x}^{2}}{{y}^{2}}{{z}^{5}}\] .
Let us first find the degree of the very first term. We shall add the numbers 3 and 3, as they are the exponents of the two variables ‘x’ and ‘y’ respectively in the first term to find the degree of the first term. The sum of 3 and 3 is 6. So, the degree of the first term is 6.
Now, we shall find the degree of the second term. We shall add the numbers 3 and 2, as they are the exponents of the two variables ‘x’ and ‘y’ respectively in the second term to find the degree of the second term. The sum of 3 and 3 is 5. So, the degree of the second term is 5.
Now, let us find the degree of the third term. The degree of the third term is 3 as it is the exponent of the only variable ‘y’ in the third term.
Let us now find the degree of the fourth and last term. We shall add the numbers 2, 2, and 5, as they are the exponents of the two variables ‘x’, ‘y’, and ‘z’ in the third term to find the degree of the third term. The sum of 2, 2, and 5 is 9. So, the degree of the first term is 9.
Now, we have to take the highest degree, which is number 9. So, the degree of the whole expression or the polynomial is 9, as it is the highest degree among the four terms.