Question

question 4 of 5
select all the correct locations on the graph.
consider function f:
$f(x) = x^3 + 3x^2 - x - 3$
the graph of function g is shown. select all the zeros that functions f and g have in common.

Explanation:

Step1: Factor $f(x)$ by grouping

Group terms:

$$\begin{align*} f(x)&=(x^3 + 3x^2) + (-x - 3)\\ &=x^2(x+3) -1(x+3)\\ &=(x^2 -1)(x+3)\\ &=(x-1)(x+1)(x+3) \end{align*}$$

Step2: Find zeros of $f(x)$

Set $f(x)=0$:
$(x-1)(x+1)(x+3)=0$
Zeros: $x=1$, $x=-1$, $x=-3$

Step3: Identify zeros of $g(x)$ from graph

Graph crosses x-axis at $x=-3$, $x=-1$, $x=1$

Step4: Find common zeros

Match zeros of $f$ and $g$

Answer:

The common zeros are $x=-3$, $x=-1$, and $x=1$ (the points $(-3,0)$, $(-1,0)$, and $(1,0)$ on the graph of $g$)