Chapter 3.6, Problem 11E

a.

To determine

To find: the zeroes of polynomial $P\left(x\right)=x^4-16$

Answer to Problem 11E

The zeroes of polynomial $P\left(x\right)=x^4-16$ is $\pm 2,\pm 2i$

Explanation of Solution

Given information: The polynomial is $P\left(x\right)=x^4-16$

Calculation: For the zeroes of $P\left(x\right)$ solve the equation $P\left(x\right)=0$ for $x$

  $\begin{array}{l}P\left(x\right)=0\\ \Rightarrow x^4-16=0\\ \Rightarrow {\left(x^2\right)}^2-4^2=0\\ \left(x^2-4\right)\left(x^2+4\right)=0\\ x^2-4=0\\ x^2=4\\ \sqrt{x^2}=\sqrt{4}\\ \left|x\right|=2\\ x=\pm 2\\ x^2+4=0\\ x^2=-4\\ \sqrt{x^2}=i\sqrt{4}\\ \left|x\right|=2i\\ x=\pm 2i\end{array}$

The zeroes of the polynomial function $P\left(x\right)$ are $\pm 2,\pm 2i$

b.

To determine

To calculate: the factored form of polynomial $P\left(x\right)=x^4-16$

Answer to Problem 11E

The factored form of polynomial $P\left(x\right)=x^4-16$ is $P\left(x\right)=\left(x-2\right),\left(x+2\right),\left(x-2i\right),\left(x+2i\right)$

Explanation of Solution

Given information: The polynomial is $P\left(x\right)=x^4-16$

Calculation: According to the factor theorem if $c$ is a zero of a polynomial function $P\left(x\right)$ the binomial $\left(x-c\right)$ is a factor of this polynomial

The zeroes of the polynomial is expressed as

  $P\left(x\right)=\left(x-2\right),\left(x+2\right),\left(x-2i\right),\left(x+2i\right)$

Hence,

The factored form of polynomial $P\left(x\right)$ is

  $P\left(x\right)=\left(x-2\right)\left(x+2\right)\left(x-2i\right)\left(x+2i\right)$