Solving $x^3 + 5x^2 + 12x + 10 \geq 3x^2 - 7x - 2$ In Interval Notation

Solving polynomial inequalities involves finding the range of values for the variable that satisfy the inequality. In this article, we will walk through the steps to solve the polynomial inequality $x^3 + 5x^2 + 12x + 10 \geq 3x^2 - 7x - 2$. This problem falls under the category of mathematics, specifically algebra, and requires a good understanding of polynomial manipulation and inequality solving techniques. Understanding the nuances of solving these kinds of inequalities is crucial for various applications in mathematics and related fields.

1. Rearrange the Inequality

The first step in solving the inequality is to move all terms to one side, setting the expression greater than or equal to zero. This rearrangement helps us to consolidate the polynomial and identify the critical points more easily. To begin, subtract $3x^2$ from both sides of the inequality:

$x^3 + 5x^2 + 12x + 10 - 3x^2 \geq -7x - 2$

Next, add $7x$ to both sides:

$x^3 + 5x^2 - 3x^2 + 12x + 7x + 10 \geq -2$

Finally, add $2$ to both sides:

$x^3 + 5x^2 - 3x^2 + 12x + 7x + 10 + 2 \geq 0$

Simplify the inequality by combining like terms:

$x^3 + 2x^2 + 19x + 12 \geq 0$

This simplified form allows us to focus on the polynomial $x^3 + 2x^2 + 19x + 12$ and find its roots, which will help determine the intervals where the inequality holds true. Rearranging the inequality is a crucial step in solving polynomial inequalities, as it sets the stage for finding critical points and testing intervals.

2. Find the Roots of the Polynomial

Now that we have the polynomial $x^3 + 2x^2 + 19x + 12 \geq 0$, we need to find its roots. These roots are the values of $x$ for which the polynomial equals zero. Finding the roots can be done through various methods, including factoring, synthetic division, or using numerical methods. In this case, we will attempt to find a rational root using the Rational Root Theorem. This theorem states that if a polynomial has a rational root $p/q$, then $p$ must be a factor of the constant term and $q$ must be a factor of the leading coefficient. For our polynomial, the constant term is $12$ and the leading coefficient is $1$. Thus, the possible rational roots are the factors of $12$, which are $\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$.

Let's test these possible roots by plugging them into the polynomial. We are looking for a value of $x$ that makes the polynomial equal to zero.

Testing $x = -1$:

$(-1)^3 + 2(-1)^2 + 19(-1) + 12 = -1 + 2 - 19 + 12 = -6$

Testing $x = -2$:

$(-2)^3 + 2(-2)^2 + 19(-2) + 12 = -8 + 8 - 38 + 12 = -26$

Testing $x = -3$:

$(-3)^3 + 2(-3)^2 + 19(-3) + 12 = -27 + 18 - 57 + 12 = -54$

We can also use synthetic division to test these roots more efficiently. Let's try synthetic division with $x = -1$:

-1 | 1   2   19   12
   |     -1  -1  -18
   ----------------
     1   1   18   -6

The remainder is $-6$, so $x = -1$ is not a root.

Let's try $x = -4$:

$(-4)^3 + 2(-4)^2 + 19(-4) + 12 = -64 + 32 - 76 + 12 = -96$

Let's try synthetic division with $x = -0.7$ (approximately -7/10):

After testing various rational roots, we find that none of the simple integer values make the polynomial equal to zero. We might need to resort to numerical methods or graphing to approximate the roots. However, we can observe that for large negative values of $x$, the $x^3$ term will dominate, making the polynomial negative. For large positive values of $x$, the $x^3$ term will dominate, making the polynomial positive. This suggests there is at least one real root. We can use a calculator or computer algebra system to find a more accurate approximation of the roots.

Using a calculator or a computer algebra system (CAS), we find that one real root is approximately $x \approx -0.702$. The other two roots are complex, which means they will not affect the intervals we are testing for the inequality. Finding the roots is a critical step, as they serve as the boundaries for the intervals we need to test.

3. Create Intervals and Test Values

Once we have found the real root(s), we can create intervals based on these roots. In our case, we have one real root, approximately $x \approx -0.702$. This root divides the number line into two intervals: $(-\infty, -0.702)$ and $(-0.702, \infty)$. To determine whether the inequality $x^3 + 2x^2 + 19x + 12 \geq 0$ holds true in each interval, we pick a test value within each interval and plug it into the polynomial.

For the interval $(-\infty, -0.702)$, let's choose a test value of $x = -1$. Plug this value into the polynomial:

$(-1)^3 + 2(-1)^2 + 19(-1) + 12 = -1 + 2 - 19 + 12 = -6$

Since $-6$ is not greater than or equal to $0$, the inequality does not hold in the interval $(-\infty, -0.702)$.

For the interval $(-0.702, \infty)$, let's choose a test value of $x = 0$. Plug this value into the polynomial:

$(0)^3 + 2(0)^2 + 19(0) + 12 = 0 + 0 + 0 + 12 = 12$

Since $12$ is greater than or equal to $0$, the inequality holds in the interval $(-0.702, \infty)$.

We also need to check the root itself, $x \approx -0.702$. Since the inequality is greater than or equal to, the root is included in the solution. Therefore, the inequality holds for $x = -0.702$.

Creating intervals and testing values is a vital step to determine where the polynomial inequality is satisfied. By testing values within these intervals, we can determine the solution set.

4. Write the Solution in Interval Notation

Based on our testing, the inequality $x^3 + 2x^2 + 19x + 12 \geq 0$ holds true for the interval $(-0.702, \infty)$ and the root $x \approx -0.702$. Since the inequality includes the "equal to" condition, we include the root in our solution using a square bracket. Thus, the solution in interval notation is:

$[-0.702, \infty)$

This notation means that all values of $x$ greater than or equal to $-0.702$ satisfy the inequality. Writing the solution in interval notation is the final step, clearly expressing the range of values that satisfy the given inequality.

In conclusion, to solve the polynomial inequality $x^3 + 5x^2 + 12x + 10 \geq 3x^2 - 7x - 2$, we rearranged the inequality, found the roots of the polynomial (approximately $x \approx -0.702$), created intervals based on the roots, tested values within each interval, and wrote the solution in interval notation. The solution to the inequality is $[-0.702, \infty)$. This methodical approach ensures that we accurately determine the solution set for polynomial inequalities. Understanding these steps is essential for solving more complex mathematical problems and real-world applications.