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.
-$$f ( x ) = - x ^ { 2 } ( x + 7 ) \left( x ^ { 2 } + 1 \right)$$
A) 0 , touches the $x$-axis and turns around;
$- 7$, crosses the $x$-axis
B) 0 , touches the $x$-axis and turns around;
7 , crosses the $x$-axis
C) 0 , touches the $x$-axis and turns around;
$- 7$, crosses the $x$-axis;
$- 1$, touches the $x$-axis and turns around
D) 0 , touches the $x$-axis and turns around;
$- 7$, crosses the x-axis;
$- 1$, crosses the $x$-axis;
1 , crosses the $x$-axis;

Question

Quizplus · Accepted Answer

The x-intercepts are found by setting $f(x) = 0$, which gives us $x = 0$ and $x = -7$. The factor $x^2$ causes the graph to touch and turn around at $x = 0$, and the linear factor $(x + 7)$ means the graph crosses the x-axis at $x = -7$. The factor $(x^2 + 1)$ does not produce real x-intercepts because $x^2 + 1 > 0$ for all real numbers $x$.

Find the X-Intercepts of the Polynomial Function f(x)=−x2(x+7)(x2+1)f ( x ) = - x ^ { 2 } ( x + 7 ) \left( x ^ { 2 } + 1 \right)f(x)=−x2(x+7)(x2+1)

Find the X-Intercepts of the Polynomial Function $f ( x ) = - x ^ { 2 } ( x + 7 ) \left( x ^ { 2 } + 1 \right)$