Unformatted text preview: Objectives Section 6.5
Rational Root Theorem! Use the rational, irrational, and imaginary root theorems to locate all the roots of a all the given polynomial equation given Rational Root Theorem
This only works if we have integer coefficients. coefficients. Possible rational roots = Find the rational roots of x 3 + x 2 − 3x − 3 = 0
List the possible rational roots: possible Test using substitution or synthetic division division p q Use the rational root Use theorem to find ALL roots theorem
2x3 − x 2 + 2x −1 = 0 Use the rational root Use theorem to find ALL roots theorem
x 3 − 2 x 2 − 5 x + 10 = 0 1 The Irrational Root The Theorem Theorem If a + b is a root of a polynomial
a− b The Imaginary Root The Theorem Theorem
If the imaginary number a + bii iis a root of bs a polynomial equation with real coefficients, then the conjugate a – bi bi also is a root also Imaginary roots also occur in conjugate also occur pairs pairs equation with rational coefficients, then the conjugate is also a root the In other words, irrational roots occur in conjugate pairs. If If 3 is a root, then − 3 is a root If 3, 7 and 2 − i are roots of a polynomial with integer coefficients, find two additional roots. two Find a thirddegree polynomial equation degree with rational coefficients that has roots 3 and 1 + i What is the degree of this polynomial? What 2...
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 Fall '09
 Johnson
 Algebra, Complex number, Irrational number

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