Chapter 4.3, Problem 20AYU

In Problems 17-22, form a polynomial function $f\left(x\right)$ with real coefficients having the given degree and zeros. Answers will vary depending on the choice of the leading coefficient. Use a graphing utility to graph the function and verify the result.

Degree 6; zeros: $i$; $4-i$; $2+i$

To determine

To form: A polynomial function with real coefficients having the given degree and zeros and to use the graphing utility to graph the function and to verify the result.

Answer to Problem 20AYU

$\text{The polynomial function }f(x)=x^6-12x^5+55x^4-120x^3+139x^2-108x+85$.

Explanation of Solution

Given:

Degree $=6$, zeros $=i,4–i,2+i$.

Since the degree $=6$ there will be 6 zeros.

Since $i,4–i,2+i$ are the complex zero of the polynomial function $f\left(x\right)$ and $f\left(x\right)$ has coefficients that are real numbers Complex zero appear as conjugate pairs.

By the conjugate pair theorem $i,4–i,2+i,-i,4+i,2-i$ are the complex zeros of the polynomial function $f\left(x\right)$.

By factor theorem $f\left(c\right)=0,\left(x-c\right)=\text{ factor of }f\left(x\right)$.

The factored form $=f\left(x\right)$

$=a\left[x-i\right]\left[x-\left(-i\right)\right]\left[x-\left(4-i\right)\right]\left[x-\left(4+i\right)\right]\left[x-\left(2+i\right)\right]\left[x-\left(2-i\right)\right]$.

Let the leading coefficient $a=1$.

$f\left(x\right)=\left[x-i\right]\left[x-\left(-i\right)\right]\left[x-\left(4-i\right)\right]\left[x-\left(4+i\right)\right]\left[x-\left(2+i\right)\right]\left[x-\left(2-i\right)\right]$

$f\left(x\right)=\left[x-i\right]\left[x+i\right]\left[\left(x-4\right)+i\right]\left[\left(x-4\right)-i\right]\left[\left(x-2\right)-i\right]\left[\left(x-2\right)+i\right]$

$f\left(x\right)=\left(x^2-i^2\right)\left[{\left(x-4\right)}^2-i^2\right][\left({\left(x-2\right)}^2-i^2\right]$

$i^2=-1$

$=(x^2+1)\left(x^2-8x+16+1\right)(x^2-4x+4+1)$

$=(x^2+1)\left(x^2-8x+17\right)\left(x^2-4x+5\right)$

$=x^6-12x^5+55x^4-120x^3+139x^2-108x+85$

Degree $=6$ at most turning points $=54$.

$Y\text{-Intercept }=f\left(x\right)=0=85$