Question

How do you solve the polynomial inequality and state the answer in interval notation given `x^6+x^3>=6`?

Answer 1

The inequality is Quadratic in form.

Explanation:

Step 1: We require zero on one side.
`x^6 + x^3 - 6 ge 0`
Step 2: Since the left side consists of a constant term, a middle term, and a term whose exponent is exactly double that on the middle term, this equation is quadratic "in form." We either factor it like a quadratic, or we use the Quadratic Formula. In this case we are able to factor.

Just as `y^2 + y - 6 = (y + 3)(y - 2)`, we now have
`x^6 + x^3 - 6 = (x^3 + 3)(x^3 - 2)`.
We treat `x^3` as though it were a simple variable, y.
If it is more helpful, you may substitute `y = x^3`, then solve for y, and finally substitute back into x.

Step 3: Set each factor equal to zero separately, and solve the equation `x^6 + x^3 - 6 = 0`. We find where the left side equals zero because these values will be the boundaries of our inequality.

`x^3 + 3 = 0`
`x^3 = -3`
`x = -root(3)3`

`x^3 -2 = 0`
`x^3 = -2`
`x = root(3)2`

These are the two real roots of the equation.
They separate the real line into three intervals:
`(-oo, -root(3)3); (-root(3)3, root(3)2); and (root(3)2, oo)`.

Step 4: Determine the sign of the left side of the inequality on each of the above intervals.

Using test points is the usual method. Select a value from each interval, and substitute it for x in the left side of the inequality. We might choose -2, then 0, and then 2.
You will discover that the Left Hand Side is
positive on `(-oo, -root(3)3)`;
negative on `(-root(3)3, root(3)2)`;
and positive on `(root(3)2, oo)`.

Step 5: Complete the problem.
We are interested in knowing where `x^6 + x^3 - 6 ge 0`.
We know now where the left side equals 0, and we know where it is positive. Write this information in interval form as:

`(-oo, -root(3)3] uu [root(3)2, oo)`.

NOTE: We have the brackets because the two sides of the inequality are equal at those points, and the original problem requires for us to include those values. Had the problem used `>` instead of `ge`, we would have used parentheses.