solve the polynomial ineqaality. express the solution set in interval notation.

332

ss33

x^3+x^2+64x+64>0

x^2(x+1) + 64(x+1) > 0
(x+1)(x^2 + 64) > 0

since x^2 + 64 is always positive
we just have to look at
x+1 > 0
x > -1

see
http://www.wolframalpha.com/input/?i=x%5E3%2Bx%5E2%2B64x%2B64

notice the graph is > 0 (above the x-axis) for all values of x , x >-1
The first graph looks like a straight line, but it isn't.
the scaling distorts the graph.

To solve the polynomial inequality x^3 + x^2 + 64x + 64 > 0, we need to find the values of x that make the expression greater than zero. Here's the step-by-step solution:

Step 1: Factor the polynomial (if possible) or find its roots.
Unfortunately, the given polynomial cannot be easily factored. So, we'll resort to finding its roots using numerical methods or graphing calculators.

Step 2: Find the critical points.
The critical points are the values of x that make the expression equal to zero. In this case, we'll find the roots of the polynomial.

Step 3: Test intervals.
Divide the number line based on the critical points and choose a test value from each interval to check if it satisfies the inequality.

Step 4: Determine the solution set.
Based on the results from step 3, determine the intervals where the expression is greater than zero.

Since I cannot directly find the roots of the given polynomial here, I recommend using a graphing calculator or a computer algebra system to solve it or get approximate values of the roots. Once you have the roots, follow steps 3 and 4 to get the solution set in interval notation.