2022-01-31

Write a polynomial function f of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. Write the polynomial in standard form. 2, 5 + i

alenahelenash

Expert

$f\left(x\right)=\left(x-2\right)\left[x-\left(5-i\right)\right]\left[x-\left(5+i\right)\right]$

$=\left(x-2\right)\left[\left(x-5\right)+i\right]\left[\left(x-5\right)-i\right)$

$=\left(x-2\right)\left[\left(x-5{\right)}^{2}-{i}^{2}\right]$

$=\left(x-2\right)\left({x}^{2}-10x+25+1\right)$

$=\left(x-2\right)\left({x}^{2}-10x+26\right)$

$={x}^{3}-10{x}^{2}+26x-2{x}^{2}+20x-52$

$={x}^{3}-12{x}^{2}+46x-52$

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