2) Yes, 6 is a polynomial. By definition, a polynomial is formed by the sum of terms of form ax^n, where "a" is any number and "n" is a natural number. In the case of 6, it is a polynomial of form 6x^0 of degree zero.
3) 9m^5 is a monomial. By definition, the degree of a monomial is the sum of the exponents of its variables. Therefore, the degree and the leading coefficient are:
Degree of the polynomial: 5
Leading coefficient of the polynomial: 9
4) When we order the polynomial 2-6y so the exponents decrease from left to right, we have:
-6y + 2
The variable "y" has exponent 1 and 2 has this form: 2y^0 (2y^0=2x1=2), so the polynomial is of degree 1.
Degree of the polynomial= 1
The leading coefficient is the coefficient of the monomial that has the highest exponent:
Leading coefficient of the polynomial = -6
5) The polynomial 2x^2y^2-8xy is already ordered, so the exponents decrease from left to right.
When it has two variables, we found the degree my adding the exponents of those variables:
Degree of the polynomial= 4
Leading coefficient of the polynomial= 2
6) The polynomial 5n^3+2n-7 is already ordered, because the exponent of the first term is 3, the exponent of the second term is 1 and the last term has exponent zero (7x^0=7x1=7).
Degree of the polynomial= 3
Leading coefficient of the polynomial= 5 (The coefficient of n^3)
7) The polynomial 5z+2z^3+3z^4 is ordered as show below:
3z^4+2z^3+5z
So, we have that degree of the polynomial and the leading coefficient are:
Degree of the polynomial= 4
Leading coefficient of the polynomial= 3 (The coefficient of z^4)
8)-2h^2+2h^4-h^6
When we order it, we have it in the following form:
-h^6+2h^4-2h^2
Let's proceed to find the degree of the polynomial and its leading coefficient, which are shown below:
Degree of the polynomial= 6
Leading coefficient of the polynomial= -1
9) The answer is: C)4
To find the degree of the polynomial -4x^3+6x^4-1 , we must find the the term with the highest exponent, and this exponent will be the degree.
As we can see, the exponent of the term 6x^4 is higher than the exponent 3 and than the exponent 0 (1x^0=1x1=1). The exponent 4 is the highest, so the degree of the polynomial is 4.
10)We already know that a monomial is a polynomial that is formed by one term. The Polynomical Function has as principal condition that the exponent of the terms have to be positive. Therefore, 3s^-2 is not a monomial, because its exponent -2 is not positive.
The answer is: D) 3s^-2
11) -4^x is not a polynomial. We can know this because, by definition, the terms of a Polynomial Function have the form ax^n, where "n" is a natural number. As we can see, the exponent of the expression -4^x is "x" , and it is not a natural number. So, -4^x does not have degree.
12) w^-3+1 is not a polynomial, because it has a negative exponent (-3). A Polynomial Funcion has the following condition: The exponent of the variable in a term always have to be a positive number. In other words, a term of a polynomial has the form ax^n, and "n" have to be a natural number.
13) 3x-5 is a polynomial because its terms have the form ax^n and its exponents are positive.
Classification: It is a binomial. We can know this because the number of its terms. By definition, a binomial is a polinomial that has two terms.
Degree of the polynomial: The highest exponent of the terms is 1, so its degree is: Degree 1.
14) 4/5f^2-1/2f+2/3 is a polynomial because the terms have the form ax^n and its exponents are positive, as is established in the definition of a Polynomial Function.
Classification: To classify this polynomial, we must count the number of its terms. As we can see, is has three terms, so it is trinomial, that is define as a polynomial that has three terms.
Degree of the polynomial: The highest exponent of the polynomial is 2, so its degree is: Degree 2.
15) 6-n^2+5n^3 is clearly a polynomial, we can know this because, as we saw before, this expression cumplies with the definition of a Polynomial Function.
Classification: When we count the number of the terms of this polynomial, we can see that it has three terms, and we already know that a polynomial that has three terms is known as a trinomial.
Degree of the polynomial: As we can see, the highest exponent of the polynomial is 3, so its degree is: Degree 3.
16)10y^4-3y^2+11 is a polynomial cumplies with the definition of a Polynomial Function and the exponents of each term are positive. So we can classify it and find its degree.
Classification: By counting the number of the terms of this polynomial, we can conclude that it has three terms, so it is definitely a trinomial (a polynomial that has three terms).
Degree of the polynomial: As we can notice, the highest exponent of the polynomial is 4, so its degree is: Degree 4.