Solve the polynomial inequality. Write the answer in interval notation. $$x ^ { 3 } - x ^ { 2 } \leq 56 x$$
A) $( - \infty , - 7 ) \cup ( 0,8 )$
B) $( - 7,8 )$
C) $( - \infty , - 7 ) \cup ( 8 , \infty )$
D) $( - 7,0 ) \cup ( 8 , \infty )$

Question

Quizplus · Accepted Answer

First, rearrange the inequality to get all terms on one side: $x^3 - x^2 - 56x \leq 0$. Factor out an $x$: $x(x^2 - x - 56) \leq 0$. Factoring the quadratic gives $x(x+7)(x-8) \leq 0$. The roots are $x = 0$, $x = -7$, and $x = 8$. Testing intervals around these roots shows that the inequality is satisfied for $x \in ( - \infty , - 7 ) \cup ( 0,8 )$.

Solve the Polynomial Inequality x3−x2≤56xx ^ { 3 } - x ^ { 2 } \leq 56 xx3−x2≤56x

Solve the Polynomial Inequality $x ^ { 3 } - x ^ { 2 } \leq 56 x$