Question

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, 1 + i, 2 - sqrt{3} \\) \\( f(x) = \square \\)

Explanation:

Step1: Identify conjugate zeros

For rational coefficients, complex zeros come in conjugate pairs: $1-i$; irrational zeros come in conjugate pairs: $2+\sqrt{3}$. All zeros: $2, 1+i, 1-i, 2-\sqrt{3}, 2+\sqrt{3}$.

Step2: Write factor form

$f(x)=(x-2)(x-(1+i))(x-(1-i))(x-(2-\sqrt{3}))(x-(2+\sqrt{3}))$

Step3: Multiply complex conjugate factors

$(x-(1+i))(x-(1-i))=(x-1)^2 - i^2 = x^2-2x+1+1=x^2-2x+2$

Step4: Multiply irrational conjugate factors

$(x-(2-\sqrt{3}))(x-(2+\sqrt{3}))=(x-2)^2 - (\sqrt{3})^2=x^2-4x+4-3=x^2-4x+1$

Step5: Multiply $(x-2)$ with $x^2-2x+2$

$(x-2)(x^2-2x+2)=x^3-2x^2+2x-2x^2+4x-4=x^3-4x^2+6x-4$

Step6: Multiply result with $x^2-4x+1$

$$\begin{align*} &(x^3-4x^2+6x-4)(x^2-4x+1)\\ =&x^5-4x^4+x^3-4x^4+16x^3-4x^2+6x^3-24x^2+6x-4x^2+16x-4\\ =&x^5-8x^4+23x^3-32x^2+22x-4 \end{align*}$$

Answer:

$x^5-8x^4+23x^3-32x^2+22x-4$