How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 5, 2+3i?

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How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 5, 2+3i?

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Answer 1 · 2018-08-05T09:36:44

$p (x) = x^{3} - 9 x^{2} + 33 x - 65$ $p(x)=x^3-9x^2+33x-65$

Explanation:

$by the factor theorem$ $"by the factor theorem"$

$if x = a is a zero then$ $"if "x=a" is a zero then "$

$(x - a) is a factor of the polynomial$ $(x-a)" is a factor of the polynomial"$

$note that complex zeros occur in conjugate pairs$ $"note that complex zeros occur in conjugate pairs"$

$x = 2 + 3 i is a zero then 2 - 3 i is also a zero$ $x=2+3i" is a zero then "2color(red)(-)3i" is also a zero"$

$p (x) = (x - 5) (x - (2 + 3 i)) (x - (2 - 3 i))$ $p(x)=(x-5)(x-(2+3i))(x-(2-3i))$

$p (x) = (x - 5) (x - 2 - 3 i) (x - 2 + 3 i)$ $color(white)(p(x))=(x-5)(x-2-3i)(x-2+3i)$

$p (x) = (x - 5) ({(x - 2)}^{2} - {(3 i)}^{2})$ $color(white)(p(x))=(x-5)((x-2)^2-(3i)^2)$

$p (x) = (x - 5) (x^{2} - 4 x + 4 + 9)$ $color(white)(p(x))=(x-5)(x^2-4x+4+9)$

$p (x) = (x - 5) (x^{2} - 4 x + 13)$ $color(white)(p(x))=(x-5)(x^2-4x+13)$

$p (x) = x^{3} - 9 x^{2} + 33 x - 65$ $color(white)(p(x))=x^3-9x^2+33x-65$

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How do you write a polynomial function of least degree and leading coefficient 1 when the zeros are 5, 2+3i?

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