How do you write a polynomial function of least degree that has real coefficients, the following given zeros 3-i,5i and a leading coefficient of 1?

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How do you write a polynomial function of least degree that has real coefficients, the following given zeros 3-i,5i and a leading coefficient of 1?

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Answer 1 · 2016-12-10T18:58:57

Any polynomial with real coefficients and imaginary or complex zeros, must have zeros that are conjugate pairs, therefore, the polynomial must have the following factors:

$y = (x - 3 + i) (x - 3 - i) (x - 5 i) (x + 5 i)$ $y = (x - 3 + i)(x - 3 - i)(x - 5i)(x + 5i)$

Multiplying the last two factors is easy; it is the sum of two squares:

$y = (x - 3 + i) (x - 3 - i) (x^{2} + 25)$ $y = (x - 3 + i)(x - 3 - i)(x^2 + 25)$

Multiplying the first two factors is a bit more difficult:

$y = (x^{2} - 3 x - i x - 3 x + 9 + 3 i + i x - 3 i - i^{2}) (x^{2} + 25)$ $y = (x^2 - 3x - ix -3x + 9 + 3i + ix -3i - i^2)(x^2 + 25)$

Combine like terms:

$y = (x^{2} - 6 x + 9 - i^{2}) (x^{2} + 25)$ $y = (x^2 - 6x + 9 - i^2)(x^2 + 25)$

Use the identity $i^{2} = - 1$ $i^2 = -1$ :

$y = (x^{2} - 6 x + 10) (x^{2} + 25)$ $y = (x^2 - 6x +10)(x^2 + 25)$

Multiply the remaining factors:

$y = x^{4} - 6 x^{3} + 10 x^{2} + 25 x^{2} - 150 x + 250$ $y = x^4 - 6x^3 +10x^2 + 25x^2 -150x + 250$

Combine like terms:

$y = x^{4} - 6 x^{3} + 35 x^{2} - 150 x + 250$ $y = x^4 - 6x^3 + 35x^2 -150x + 250$

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How do you write a polynomial function of least degree that has real coefficients, the following given zeros 3-i,5i and a leading coefficient of 1?

1 Answer

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