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Diane M. asked • 11/14/17

Find a degree 3 polynomial with real coefficients having zeros 5 and 5 - 5i and a lead coefficient of 1. Write P in expanded form.

Find a degree 3 polynomial with real coefficients having zeros

5 and 5 - 5i and a lead coefficient of 1. Write P in expanded form.

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Mark M. answered • 11/14/17

TUTOR 5.0 (243)

Mathematics Teacher - NCLB Highly Qualified

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P(x) = (x - 5)(x - (5 - 5i))(x - (5 + 5i))

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Andrew M. answered • 11/15/17

TUTOR New to Wyzant

Mathematics - Algebra a Specialty / F.I.T. Grad - B.S. w/Honors

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A zero at 5 means (x-5) is a factor

A zero at 5-5i means (x - (5-5i)) or (x-5+5i) is a factor

If 5-5i is a factor then so is the complex conjugate 5+5i:

A zero at 5+5i means (x-(5+5i)) or (x-5-5i) is a factor

p(x) = a(x-5)(x-5+5i)(x-5-5i)

Since the leading coefficient is 1, a=1

p(x) = (x-5)(x-5+5i)(x-5-5i)

p(x) = (x2 - 5x + 5ix - 5x + 25 - 25i)(x-5-5i)

p(x) = (x2 - 10x + 25 + 5ix - 25i)(x-5-5i)

multiplying through by each term and adding like terms gives us:

p(x) = x3 - 5x2- 5ix2

- 10x2 + 50x + 50ix

+ 25x - 125 - 125i

+5ix2 + 25x - 25ix

- 25ix - 125 +125i

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p(x) = x3 - 15x2 + 100x - 250