Consider the following polynomial.
-2a^3 b^2 + 5ab^5 + 7b^4 + 8
Answer Choices:
The degree of the polynomial is
(5, 6, 15)
The leading coefficient is:
Answer Choices:
(-2, 5, 7, 8)
The constant term is:
Answer Choices:
(-2, 5, 7, 8)
The leading term is
Answer Choices:
(-2a^3 b^2
5ab^5
7b^4
8)
(-2, 5, 7, 8)
The degree of the polynomial is the highest degree of any term in the polynomial. In this case, the highest degree is 5 for the term 5ab^5. Therefore, the degree of the polynomial is 5.
The leading coefficient is the coefficient of the term with the highest degree. In this case, the coefficient of the term 5ab^5 is 5. Therefore, the leading coefficient is 5.
The constant term is the term that does not have any variables. In this case, the constant term is 8. Therefore, the constant term is 8.
The leading term is the term with the highest degree. In this case, the term with the highest degree is 5ab^5. Therefore, the leading term is 5ab^5.
are you sure?
Apologies for the confusion. Let's go through the analysis again:
The degree of a polynomial is the highest sum of the exponents of all the terms. In this case, the highest sum of exponents is 5 (for the term 5ab^5), so the degree of the polynomial is 5.
The leading coefficient is the coefficient of the term with the highest degree. In this case, the highest degree term is 5ab^5, and its coefficient is 5. Therefore, the leading coefficient is 5.
The constant term is the term that does not have any variables. In this polynomial, the term 8 does not have any variables. Hence, the constant term is 8.
The leading term is the term with the highest degree. In this case, it is 5ab^5.
To summarize:
- The degree of the polynomial is 5.
- The leading coefficient is 5.
- The constant term is 8.
- The leading term is 5ab^5.
but isnt the degree of the poloynomial 6?
You are correct, my previous response contained an error. I apologize for the confusion.
Looking at the given polynomial -2a^3 b^2 + 5ab^5 + 7b^4 + 8, the term with the highest degree is 5ab^5, which has a degree of 6 (3+5=8). Therefore, the correct degree of the polynomial is 6.
To summarize the correct answers:
- The degree of the polynomial is 6.
- The leading coefficient is 5.
- The constant term is 8.
- The leading term is 5ab^5.
ok but how do you get these answers
To determine the degree of a polynomial, you need to find the term with the highest degree, which is the largest sum of the exponents on the variables in the term. In this case, the term with the highest degree is 5ab^5, where the sum of the exponents (3+5) is 8. Therefore, the degree of the polynomial is 6.
The leading coefficient is the coefficient of the term with the highest degree. In this case, the highest degree term is 5ab^5, and its coefficient is 5. Therefore, the leading coefficient is 5.
The constant term is the term that does not contain any variables. In this polynomial, the term 8 does not contain any variables, making it the constant term.
The leading term is simply the term with the highest degree, which, as mentioned earlier, is 5ab^5.
Therefore, by analyzing the exponents of the variables, coefficients, and constants in the polynomial, we can determine the degree, leading coefficient, constant term, and leading term.