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5.5 – Apply the Remainder and Factor Theorems

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. 5.5 – Apply the Remainder and Factor Theorems. Example 6:

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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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