How to Solve a Cubic Inequality | Summary and Q&A

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April 20, 2022
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The Math Sorcerer
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How to Solve a Cubic Inequality

TL;DR

This video explains how to solve an inequality by factoring and synthetic division, using the example of x³ + x² - 5x + 3 ≤ 0.

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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 non-negative, 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).

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