Question

Determine if the function is a polynomial function. If the function is a polynomial function, state the degree and leading coefficient. If the function is not a polynomial, state why. f(x)=x^4(2-x^3)+1

Answer 1

The correct statement about the function is: D. This is a polynomial function of degree 7 with a leading coefficient of -1.

In Mathematics and Euclidean Geometry, the degree of a polynomial function is sometimes referred to as an absolute degree and it represents the leading coefficient of each of its term.

Based on the information provided above, we would evaluate the given function by rewriting in standard form as follows;

`f(x)=x^4(2-x^3)+1\\\\f(x)=2x^4-x^7+1\\\\f(x)=-x^7+2x^4+1`

In this context, we can logically deduce that the function is a polynomial function. Therefore, the degree and leading coefficient of this polynomial function are as follows;

Degree = 7 (The highest power of x).

Leading coefficient = -1 (The coefficient of the term with the highest power of x).

Answer 2

The correct option is

This is a polynomial function of degree 7 with a leading coefficient of -1

Explanation:

Functions that consist of a variable such as x raised to positive integer powers which are equal to or larger than zero added together to make the function are known as polynomial functions

Therefore, the function in the question which is f(X) = x⁴ × (2 - x³) + 1

Which can be expanded as follows

f(x) = 2·x⁴ - x⁷ + 1, which is the same as given as follow equation;

f(x) = -x⁷ + 2·x⁴ + 1

Which is polynomial function with a leading coefficient of -1 as it consists of only whole number positive powers of x including the powers of x 4 and 7

Therefore, the correct option is that f(x) is a polynomial function of degree 7 with a leading coefficient of -1.