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