Question

writing a polynomial function of a graph
write the 4th degree polynomial function shown by the graph.
the factors of the polynomial functions are
(x + 3)(x − 1)(x + 2)(x − 3)
(x + 3)(x + 1)(x − 2)(x − 3)
(x + 3)(x + 1)(x − 2)(x − 3)(x − 36)
(x + 3)(x + 1)(x − 2)(x − 3)(x − 0)

Explanation:

Step1: Identify x-intercepts

The graph crosses the x - axis at \(x=-3\), \(x = - 1\), \(x = 2\), \(x=3\). For a polynomial, if \(r\) is a root (x - intercept), then \((x - r)\) is a factor. So the factors corresponding to these roots are \((x+3)\) (since \(r=-3\), \(x-(-3)=x + 3\)), \((x + 1)\) (since \(r=-1\), \(x-(-1)=x + 1\)), \((x-2)\) (since \(r = 2\), \(x-2\)), and \((x - 3)\) (since \(r=3\), \(x - 3\)). So the polynomial should have factors \((x + 3)(x + 1)(x-2)(x - 3)\).

Step2: Check the degree

The polynomial is a 4th - degree polynomial. The first option \((x + 3)(x-1)(x + 2)(x - 3)\) has roots at \(x=-3\), \(x = 1\), \(x=-2\), \(x = 3\) which do not match the x - intercepts of the graph. The third option \((x + 3)(x + 1)(x-2)(x - 3)(x - 36)\) is a 5th - degree polynomial (since it has 5 factors), and the fourth option \((x + 3)(x + 1)(x-2)(x - 3)(x-0)\) is also a 5th - degree polynomial. The second option \((x + 3)(x + 1)(x-2)(x - 3)\) is a 4th - degree polynomial with the correct roots.

Answer:

\((x + 3)(x + 1)(x-2)(x - 3)\) (the second option in the list of factors)