Question

write a polynomial function ( f ) of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros. write the function in standard form.
4, ( -sqrt{5} )
( f(x) = square )

Explanation:

Step1: Identify conjugate zero

Since coefficients are rational, $\sqrt{5}$ is also a zero (conjugate of $-\sqrt{5}$).

Step2: Write factor form

Use zeros to get factors: $(x-4)$, $(x+\sqrt{5})$, $(x-\sqrt{5})$
$f(x)=(x-4)(x+\sqrt{5})(x-\sqrt{5})$

Step3: Multiply radical factors

Use difference of squares: $(x+\sqrt{5})(x-\sqrt{5})=x^2-5$
$f(x)=(x-4)(x^2-5)$

Step4: Distribute $(x-4)$

Multiply each term in $x^2-5$ by $x-4$:
$f(x)=x(x^2-5)-4(x^2-5)=x^3-5x-4x^2+20$

Step5: Rearrange to standard form

Order terms by descending degree:
$f(x)=x^3-4x^2-5x+20$

Answer:

$x^3-4x^2-5x+20$