How do you write a polynomial function with minimum degree whose zeroes are 5 and 3+2i?

Nov 11, 2017

see below

Explanation:

a polynomial$\text{ } P \left(x\right)$ with minimum degree.

We have the coefficients real, therefore all complex roots occur in conjugate pairs

we have zeros

$x = 5 \implies \left(x - 5\right)$" "is a factor of $P \left(x\right)$

$x = 3 + 3 i \implies \left(x - \left(3 + 2 i\right)\right)$a factor

$\therefore x = 3 - 2 i \implies \left(x - \left(3 - 2 i\right)\right)$a factor

$P \left(x\right) = \left(x - 5\right) \left(x - \left(3 + 2 i\right)\right) \left(x - \left(3 - 2 i\right)\right)$

$P \left(x\right) = \left(x - 5\right) \left({x}^{2} - x \left(3 - 2 i\right) - x \left(3 + 2 i\right) + \left(3 + 2 i\right) \left(3 - 2 i\right)\right)$

$= \left(x - 5\right) \left({x}^{2} - 3 x + \cancel{2 x i} - 3 x \cancel{- 2 x i} + 9 + 4\right)$

$= \left(x - 5\right) \left({x}^{2} - 6 x + 13\right)$

$= {x}^{3} - 6 {x}^{2} + 13 - 5 {x}^{2} + 30 x - 65$

${x}^{3} - 11 {x}^{2} + 30 x - 52$