Michigan State University - Diagnostic Exam

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4.19

  •   Let h (x ) = x3 - 6x2 + 21x - 26 . If 2 is a root of h(x) then the other two roots are?

    a) -2 ± 3i b) 3 ± 2i c) 2 ± 3i d) -3 ± 2i e) ±2i

Solution:

2 h(x) = (x - 2 )(x - 4x + 13 ) = (x - 2 )(x2 - 4x + 4 + 9) ( ) = (x - 2 ) (x - 2)2 + 9 = (x - 2 )(x - 2 + 3i)(x - 2 - 3i)

It follows that h(2 + 3i) = h(2 - 3i) = 0 .

Diagnostic:

If you miss this question please review the Fundamental Theorem of Algebra.

Also, see 4.5: Quadratic Equations and 4.15: Polynomial Division .

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July 15, 2008