Consider the following polynomial. -2a^3b^2 + 5ab^5 +7b^4 + 8

Question

The polynomial has a degree of ___.
The leading coefficient of the polynomial is ___.
the constant term of the polynomial is ___.
The leading term of the polynomial is ___.

Answer 1

To analyze the polynomial \(-2a^3b^2 + 5ab^5 + 7b^4 + 8\), we can determine the following:

1. Degree of the polynomial: The degree of a polynomial is the highest sum of the exponents of the variables in any single term.

   • For the term \(-2a^3b^2\), the sum of the exponents is \(3 + 2 = 5\).
   • For the term \(5ab^5\), the sum of the exponents is \(1 + 5 = 6\).
   • For the term \(7b^4\), the sum of the exponents is \(4\).
   • For the constant term \(8\), there are no variables, so the degree is \(0\).

   The highest degree among these is \(6\), so the degree of the polynomial is 6.

2. Leading coefficient of the polynomial: The leading coefficient is the coefficient of the term with the highest degree, which we identified as \(5ab^5\). The coefficient here is \(5\), so the leading coefficient is 5.

3. Constant term of the polynomial: The constant term is the term that does not contain any variables, which in this polynomial is \(8\). Thus, the constant term is 8.

4. Leading term of the polynomial: The leading term is the term with the highest degree, which we determined to be \(5ab^5\). Therefore, the leading term of the polynomial is 5ab^5.

To summarize:

• The degree of the polynomial is 6.
• The leading coefficient of the polynomial is 5.
• The constant term of the polynomial is 8.
• The leading term of the polynomial is 5ab^5.