Question

which of the following is a 5th degree polynomial with a leading coefficient of 1 and a constant term of negative 6? (hint: if there is no coefficient in front of a variable, that means the coefficient is actually 1.)

Explanation:

Step1: Recall the definition of polynomial degree and leading coefficient

The degree of a polynomial is the highest power of the variable. The leading coefficient is the coefficient of the term with the highest degree.
For a fifth - degree polynomial, the highest power of \(x\) is \(x^{5}\). The general form of a polynomial term is \(ax^{n}\), where \(a\) is the coefficient and \(n\) is the degree.

Step2: Analyze each option

• Option \(x^{5}-3x^{2}+2x - 5\):

The leading coefficient of \(x^{5}\) is \(1\) (since \(x^{5}=1\times x^{5}\)), and the constant term is \(- 5\).

• Option \(x^{4}-3x^{2}+2x - 6\):

The highest - degree term is \(x^{4}\), so it is a fourth - degree polynomial.

• Option \(x^{5}-3x^{2}+2x + 6\):

The leading coefficient of \(x^{5}\) is \(1\), and the constant term is \(6\) (not \(-6\)).

• Option \(x^{5}-3x^{2}+2x - 6\):

The leading coefficient of \(x^{5}\) is \(1\), the polynomial is of degree \(5\), and the constant term is \(-6\).

Answer:

\(x^{5}-3x^{2}+2x - 6\)