Question

write a polynomial function ( f ) of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros: ( 3, 4 + 2i, 1 + sqrt{7} ). write the polynomial in standard form. ( f(x) = square )

Explanation:

Step1: Identify conjugate zeros

For rational coefficients, irrational zero $1+\sqrt{7}$ has conjugate $1-\sqrt{7}$; complex zero $4+2i$ has conjugate $4-2i$. All zeros: $3, 4+2i, 4-2i, 1+\sqrt{7}, 1-\sqrt{7}$.

Step2: Write factor form

$f(x)=(x-3)(x-(4+2i))(x-(4-2i))(x-(1+\sqrt{7}))(x-(1-\sqrt{7}))$

Step3: Simplify complex factors

Group $(x-(4+2i))(x-(4-2i))$:

$$\begin{align*} &=(x-4-2i)(x-4+2i)\\ &=(x-4)^2-(2i)^2\\ &=x^2-8x+16+4\\ &=x^2-8x+20 \end{align*}$$

Step4: Simplify irrational factors

Group $(x-(1+\sqrt{7}))(x-(1-\sqrt{7}))$:

$$\begin{align*} &=(x-1-\sqrt{7})(x-1+\sqrt{7})\\ &=(x-1)^2-(\sqrt{7})^2\\ &=x^2-2x+1-7\\ &=x^2-2x-6 \end{align*}$$

Step5: Multiply quadratic factors

Multiply $(x^2-8x+20)(x^2-2x-6)$:

$$\begin{align*} &=x^2(x^2-2x-6)-8x(x^2-2x-6)+20(x^2-2x-6)\\ &=x^4-2x^3-6x^2-8x^3+16x^2+48x+20x^2-40x-120\\ &=x^4-10x^3+30x^2+8x-120 \end{align*}$$

Step6: Multiply by $(x-3)$

$$\begin{align*} &=(x-3)(x^4-10x^3+30x^2+8x-120)\\ &=x(x^4-10x^3+30x^2+8x-120)-3(x^4-10x^3+30x^2+8x-120)\\ &=x^5-10x^4+30x^3+8x^2-120x-3x^4+30x^3-90x^2-24x+360\\ &=x^5-13x^4+60x^3-82x^2-144x+360 \end{align*}$$

Answer:

$x^5-13x^4+60x^3-82x^2-144x+360$