Solve the polynomial inequality and graph the solution set on a real number line. Express the solution set in interval notation. [tex]x^3+4 x^2-16 x-64 \textgreater \ 0[/tex] Solve the inequality. What is the solution set? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The solution set is [ ]. (Simplify your answer. Type your answer in interval notation. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the expression.) B. The solution set is the empty set.
Answer (2)
Factor the polynomial: x 3 + 4 x 2 − 16 x − 64 = ( x − 4 ) ( x + 4 ) 2 .
Analyze the sign of 0"> ( x − 4 ) ( x + 4 ) 2 > 0 .
Determine the solution: 4"> x > 4 and x = − 4 .
Express the solution in interval notation: ( 4 , ∞ ) .
Explanation
Understanding the Problem We are given the polynomial inequality 0"> x 3 + 4 x 2 − 16 x − 64 > 0 . Our goal is to find the solution set for this inequality and express it in interval notation.
Factoring the Polynomial To solve the inequality, we first need to factor the polynomial x 3 + 4 x 2 − 16 x − 64 . We can use factoring by grouping:
x 3 + 4 x 2 − 16 x − 64 = x 2 ( x + 4 ) − 16 ( x + 4 ) = ( x 2 − 16 ) ( x + 4 ) = ( x − 4 ) ( x + 4 ) ( x + 4 ) = ( x − 4 ) ( x + 4 ) 2 .
Rewriting the Inequality Now the inequality becomes 0"> ( x − 4 ) ( x + 4 ) 2 > 0 . We need to analyze the sign of the expression ( x − 4 ) ( x + 4 ) 2 .
Analyzing the Sign Notice that the term ( x + 4 ) 2 is always non-negative. It is zero when x = − 4 . The term ( x − 4 ) is positive when 4"> x > 4 and negative when x < 4 .
Determining the Solution Therefore, 0"> ( x − 4 ) ( x + 4 ) 2 > 0 when 4"> x > 4 and x = − 4 .
Expressing the Solution The solution set in interval notation is ( 4 , ∞ ) .
Examples
Polynomial inequalities can be used to model various real-world scenarios. For example, a company might use a polynomial inequality to determine the range of production levels that will result in a profit greater than a certain amount. Similarly, in physics, polynomial inequalities can be used to describe the motion of objects or the behavior of systems under certain conditions. Understanding how to solve these inequalities is crucial for making informed decisions and predictions in these fields.
The solution set for the inequality 0"> x 3 + 4 x 2 − 16 x − 64 > 0 is ( − ∞ , − 4 ) ∪ ( 4 , ∞ ) . This means that the expression is positive in these intervals. Thus, the answer in interval notation is ( − ∞ , − 4 ) ∪ ( 4 , ∞ ) .
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