5.5 – apply the remainder and factor theorems the remainder theorem provides a quick way to find...

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:

Given that P(x) = x5 – 2x3 – x2 + 2, what is the remainder when P(x) is divided by x – 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 ?

5.5 – Apply the Remainder and Factor Theorems

5.5– Theorems About Roots of Polynomial Equations

Example 2:

What are the rational roots of

2x3 – x2 + 2x + 5 = 0

5.5– Theorems About Roots of Polynomial Equations

Example 1b:

What are the rational roots of

3x3 + 7x2 + 6x – 8 = 0

5.5– Theorems About Roots of Polynomial Equations

Example 2:

What are the rational roots of

15x3 – 32x2 + 3x + 2 = 0

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.

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?

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!!!