Chapter 4.3, Problem 35AYU

To determine

To find: The Complex zeros of the polynomial function $f\left( x\right)$ whose coefficients are real numbers and to write in factored form.

Answer to Problem 35AYU

$\text{The complex zeros }=\pm i,\pm 2i$.

$\text{Factored form of}f\left( x \right)=x^4 +5x^2 +4=\left[ x-i)\right]\left[ x+i\right]\left[ x-2i\right]\left[ x+2i\right]$.

Explanation of Solution

Given:

$f\left( x\right)=x^4+5x^2+4$

Degree of the polynomial function $f\left( x\right)=x^4+5x^2+4=4$.

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

From Descartes’ rule of signs.

$f\left( x\right)=x^4+5x^2+4$

Therefore $f\left( x\right)=x^4+5x^2+4$ has no positive real zero.

$f\left( -x\right)= x^4+5x^2+4$

Therefore $f\left( x\right)=x^4+5x^2+4$ have no negative real zero.

Let $x^2=y$.

$f\left( x\right)= x^4+5x^2+4$ can be in written $=\left( y^2 +5y+ 4\right)$

$\left( y^2 +5y+ 4\right)=$ quadratic equation $\left( a{y}^2 +by+c\right)$

$\left( y^2 +5y+4\right)=\left( y^2 +4y+y+4\right)=\left( y+4\right)\left( y+1\right)$

$y=-4, y=-1$

$x^2=-4, x^2=-1$

The complex zeros $=\pm 2i,\pm i$.

Factored form of $f\left( x\right)=x^4+5x^2+4$.

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