How to Solve a Cubic Inequality  Summary and Q&A
TL;DR
This video explains how to solve an inequality by factoring and synthetic division, using the example of x³ + x²  5x + 3 ≤ 0.
Key Insights
 😑 Factoring can help determine the zeros, factors, and remaining expressions of a polynomial.
 Synthetic division simplifies the process of factoring and finding the roots of a polynomial.
 🧑🏭 Checking if one is a zero can quickly identify a factor of the polynomial.
Questions & Answers
Q: How is factoring helpful in solving the given inequality?
Factoring helps identify the zeros of the polynomial, which are crucial in determining the intervals where the inequality holds true.
Q: Why is one always a good number to check as a zero?
One is often a good number to check as a zero because it is easy to evaluate the polynomial at one, and if it equals zero, it confirms one as a zero.
Q: How is synthetic division used in finding the factors of the polynomial?
Synthetic division is used to divide the polynomial by the factor (x  1), allowing us to find the quotient and the remaining quadratic expression.
Q: How does factoring the quadratic expression help in solving the inequality?
By factoring the quadratic expression (x  1)²(x + 3), it becomes clear that (x  1)² is always nonnegative, while (x + 3) determines the sign of the inequality.
Summary & Key Takeaways

The video demonstrates the steps for factoring the given polynomial and using synthetic division to find its roots.

By checking if one is a zero of the polynomial, it is determined that x  1 is a factor.

After dividing by x  1, the remaining polynomial is factored further into (x  1)²(x + 3).