Question

consider the polynomial function (f).

\\f(x) = -5x^4(x + 2)^4\\

without graphing the function, answer each of the following questions.

a) what is the degree of (f)?

b) what is the leading coefficient of (f)?

c) what is the maximum number of (x)-intercepts that the graph of (f) can have?

d) what is the maximum number of turning points that the graph (f) can have?

Explanation:

Determine the degree and leading coefficient of the polynomial

$$
LATEXBLOCK0
$$

The highest power of \(x\) is \(8\), so the degree is \(8\).
The coefficient of \(x^8\) is \(-5\), so the leading coefficient is \(-5\).

Determine the maximum number of x-intercepts

For a polynomial of degree \(n\), the maximum number of real roots (and thus \(x\)-intercepts) is \(n\).

$$ \text{Maximum } x\text{-intercepts} = 8 $$

Determine the maximum number of turning points

For a polynomial of degree \(n\), the maximum number of turning points is \(n - 1\).

$$ \text{Maximum turning points} = 8 - 1 = 7 $$

Answer:

Question a

The degree of \(f\) is 8.

Question b

The leading coefficient of \(f\) is -5.

Question c

The maximum number of \(x\)-intercepts that the graph of \(f\) can have is 8.

Question d

The maximum number of turning points that the graph of \(f\) can have is 7.