Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
-$$f ( x ) = ( x - 3 ) ^ { 2 } \left( x ^ { 2 } - 25 \right)$$
A) 3 , touches the $x$-axis and turns around;
$- 5$, crosses the $x$-axis;
5 , crosses the $x$-axis
B) 3 , touches the $x$-axis and turns around;
$- 5$, touches the $x$-axis and turns around;
5 , touches the $x$-axis and turns around
C) 3 , touches the $x$-axis and turns around;
25 , touches the $x$-axis and turns around
D) $- 3$, touches the $x$-axis and turns around;
25 , crosses the $x$-axis

Question

Find the x-intercepts of the polynomial function. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each intercept.
-$$f ( x ) = ( x - 3 ) ^ { 2 } \left( x ^ { 2 } - 25 \right)$$
A) 3 , touches the $x$-axis and turns around;
$- 5$, crosses the $x$-axis;
5 , crosses the $x$-axis
B) 3 , touches the $x$-axis and turns around; 
 $- 5$, touches the $x$-axis and turns around;
 5 , touches the $x$-axis and turns around
C) 3 , touches the $x$-axis and turns around;
25 , touches the $x$-axis and turns around
D) $- 3$, touches the $x$-axis and turns around; 
 25 , crosses the $x$-axis

Quizplus · Accepted Answer

The x-intercepts are found by setting $f(x) = 0$. The given function can be factored as $(x - 3)^2(x + 5)(x - 5)$. The intercept at $x = 3$ is a double root, meaning the graph touches the x-axis and turns around at this point. The intercepts at $x = -5$ and $x = 5$ are single roots, meaning the graph crosses the x-axis at these points.

Find the X-Intercepts of the Polynomial Function f(x)=(x−3)2(x2−25)f ( x ) = ( x - 3 ) ^ { 2 } \left( x ^ { 2 } - 25 \right)f(x)=(x−3)2(x2−25)

Find the X-Intercepts of the Polynomial Function $f ( x ) = ( x - 3 ) ^ { 2 } \left( x ^ { 2 } - 25 \right)$