Chapter 2, Problem 123RE

To determine

To calculate: all the zeros of the function using the given zero.

Answer to Problem 123RE

  $-3,2i,-2i$

Explanation of Solution

Given information:

Function: $f(x)=x^3+3x^2+4x+12$

Zero: $-2i$

Formula used:

To find the zeros of a quadratic equation,

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

Concept used:

Complex zeros occur in conjugate pairs.

Calculation:

The given zero is $-2i$ which is a complex number.

Since complex zeros occur in conjugate pairs, $2i$ is also a zero of the given function.

The factor form of the known zeros are $(x-2i)$ and $(x+2i)$

To find the other zeros of the given function, multiply the known factors as follows:

  $\begin{array}{c}(x-2i)(x+2i)=x^2+2xi-2xi-4i^2\\ =x^2+4[\because i^2=-1]\end{array}$

Divide the given function using the quadratic equation obtained by multiplying the known factors.

  $x^2+4\begin{array}{c}\hfill x+3\\ \hfill \overline{)\begin{array}{l}{x}^3+3x^2+4x+12\\ \frac{x^3+0x^2+4x}{\begin{array}{c}3x^2+12\\ \frac{3x^2+12}{0}\end{array}}\end{array}}\end{array}$

So, the third factor of the given function is ( $x+3$ )

Therefore,

  $f(x)=(x^2+4)(x+3)$

Conclusion:

The zeros of the given function are $-3,2i,-2i$