Zeroes of Polynomial Function and Intercepts

Polynomial Functions and Models

Some text here about the topic of discussion

​A polynomial function is a function of the form

Open Ended

Determine which of the following are polynomials. For those that are, state the degree.

$$\left(a\right)\ f\left(x\right)=3x^2-4x+5$$  

$$\left(b\right)\ g\left(x\right)=\sqrt[3]{x}\ -5$$  

$$\left(c\right)\ F\left(x\right)=\frac{3x^5}{5-2x}$$  

If f is a polynomial function and r is a real number for which   f(r)=0, then r is called a (real) zero of f, or root of f. If r is a (real) zero of f, then:

(a) r is an x-intercept of the graph  of f.

(b) (x - r) is a factor of f.

​Theorem

If f is a polynomial function of degree n, then f has at most n-1 turning points. 

Open Ended

For the polynomial function $$f\left(x\right)=\left(x+1\right)^2\left(x-5\right)\left(x+4\right)$$  

(a)  Find the x- and y-intercepts of the graph of f.

(b) Determine whether the graph crosses or touches the x-axis at each x-intercept.

(c) Determine the maximum number of turning points on the graph of f.