Chapter 4.3, Problem 28AYU

In Problems 23-30, use the given zero to find the remaining zeros of each function.

$f\left(x\right)=x^4-7x^3+14x^2-38x-60$; zero: $1+3i$

To determine

To find:The remaining zeros of the polynomial function $f\left(x\right)$ whose coefficients are real numbers.

Answer to Problem 28AYU

$\text{Factored form of }f\left(x\right)=x^4-7x^3+14x^2-38x-60=\left(x+1\right)\left[{\left(x-1\right)}^2-{\left(3i\right)}^2\right]\left(x-6\right)$.

Explanation of Solution

Given:

$f\left(x\right)=x^4-7x^3+14x^2-38x-60$

$\text{Zero }=1+3i$.

Degree of the polynomial function $f\left(x\right)=x^4-7x^3+14x^2-38x-60=4$.

Therefore $f\left(x\right)$ have 4 zeros.

From Descartes’ rule of signs.

$f\left(x\right)=x^4-7x^3+14x^2-38x-60$

$+To–-to++to-$

Therefore $f\left(x\right)=x^4-7x^3+14x^2-38x-60$ have 3 or 1 positive real zeros.

$f\left(-x\right)=x^4+7x^3+14x^2+38x-60$

$+\text{ To }-$

Therefore $f\left(x\right)=x^4-7x^3+14x^2-38x-60$ have 1 negative real zero.

Rational zeros theorem provides information about the potential rational zeros of the polynomial function with integer coefficients.

Zeros of $a_0=p=\pm 1,\pm 2,\pm 3,\pm 5,\pm 6,\pm 10,\pm 12,\pm 15,\pm 20,\pm 30,\pm 60$.

Zeros of $a_n=q=\pm 1$,

$\frac{p}{q}=\pm 1,\pm 2,\pm 3,\pm 5,\pm 6,\pm 10,\pm 12,\pm 15,\pm 20,\pm 30,\pm 60$

By Synthetic division.

The coefficients of $f\left(x\right)=1-714-38-60$.

The negative real zero $=-1$.

The depressed equation $=x^3-8x^2+22x-60$.

The complex zero $=1+3i$.

From conjugate pair theorem the complex zero occur only in complex pairs.

The complex zeros $=\left(1+3i\right)\left(1-3i\right)$.

The real zeros $=-1$.

Factored form of $f\left(x\right)=x^4-7x^3+14x^2-38x-60$

$=\left(x+1\right)\left[x-\left(1+3\text{i}\right)\right]\left[x-\left(1-3\text{i}\right)\right]\left(x-6\right)$

$f\left(x\right)=\left(x+1\right)\left[\left(x-1\right)-3i\right]\left[\left(x-1\right)+3i\right]\left(x-6\right)$

$f\left(x\right)=\left(x+1\right)\left[{\left(x-1\right)}^2-{\left(3i\right)}^2\right]\left(x-6\right)$