Identify all intercepts of $f ( x ) = \frac { x ^ { 2 } + 7 x + 10 } { x ^ { 3 } + 5 x ^ { 2 } - x - 5 }$ .
A)x-intercept: none; y-intercept: none
B)x-intercept: $\left( - \frac { 7 } { 5 } , 0 \right)$ ; y-intercept: none
C)x-intercept: (-2, 0); y-intercept: (0, -2)
D)x-intercepts: (-5, 0), (-1,0), (1, 0); y-intercept: (0, -2)
E)x-intercept: (0, 0); y-intercept: (0, 0)

Question

Quizplus · Accepted Answer

To find the x-intercepts, set $f(x) = 0$, which gives the roots of the numerator as long as they are not roots of the denominator. The numerator $x^2 + 7x + 10$ factors to $(x+2)(x+5)$, but $x+5$ is a factor of the denominator, leaving $x = -2$ as the only valid x-intercept. For the y-intercept, set $x = 0$, giving $f(0) = \frac{10}{-5} = -2$, so the y-intercept is (0, -2).

Identify All Intercepts Of f(x)=x2+7x+10x3+5x2−x−5f ( x ) = \frac { x ^ { 2 } + 7 x + 10 } { x ^ { 3 } + 5 x ^ { 2 } - x - 5 }f(x)=x3+5x2−x−5x2+7x+10​

Identify All Intercepts Of $f ( x ) = \frac { x ^ { 2 } + 7 x + 10 } { x ^ { 3 } + 5 x ^ { 2 } - x - 5 }$