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5.5 – Apply the Remainder and Factor Theorems

Learn about the Remainder and Factor Theorems, solve polynomial division problems, find rational roots of equations, and understand Descartes' Rule of Signs. Explore examples and graphical confirmations of real and complex roots.

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5.5 – Apply the Remainder and Factor Theorems

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  1. 5.5 – Apply the Remainder and Factor Theorems The Remainder Theorem provides a quick way to find the remainder of a polynomial long division problem.

  2. 5.5 – Apply the Remainder and Factor Theorems Example 6: Given that P(x) = x5 – 2x3 – x2 + 2, what is the remainder when P(x) is divided by x – 3?

  3. 5.5 – Apply the Remainder and Factor Theorems Example 6b: Given that P(x) = x5 – 3x4 - 28x3+ 5x + 20, what is the remainder when P(x) is divided by x + 4 ?

  4. 5.5 – Apply the Remainder and Factor Theorems

  5. 5.5– Theorems About Roots of Polynomial Equations Example 2: What are the rational roots of 2x3 – x2 + 2x + 5 = 0

  6. 5.5– Theorems About Roots of Polynomial Equations Example 1b: What are the rational roots of 3x3 + 7x2 + 6x – 8 = 0

  7. 5.5– Theorems About Roots of Polynomial Equations Example 2: What are the rational roots of 15x3 – 32x2 + 3x + 2 = 0

  8. 5.5– Theorems About Roots of Polynomial Equations The French mathematician René Descartes (1596 – 1650) recognized a connection between the roots of a polynomial equation and the + and – signs in standard form.

  9. 5.5– Theorems About Roots of Polynomial Equations Example 3: What does Descartes’ Rule of Signs tell you about the real roots of x3 – x2 + 1 = 0?

  10. 5.5– Theorems About Roots of Polynomial Equations Example 3b: What does Descartes’ Rule of Signs tell you about the real roots of 2x4 – x3 + 3x2 – 1 = 0? Can you confirm real and complex roots graphically? Explain!!!

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