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) = x 5 – 2x 3 – x 2 + 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) = x 5 – 3x 4 - 28x 3 + 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 2x 3 – x 2 + 2x + 5 = 0

6. 5.5– Theorems About Roots of Polynomial Equations Example 1b: What are the rational roots of 3x 3 + 7x 2 + 6x – 8 = 0

7. 5.5– Theorems About Roots of Polynomial Equations Example 2: What are the rational roots of 15x 3 – 32x 2 + 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 x 3 – x 2 + 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 2x 4 – x 3 + 3x 2 – 1 = 0? Can you confirm real and complex roots graphically? Explain!!!