 # Find the complex zeros of the following polynomial function. Write f in factored form. x^4+2x^3+22x^2+50x-75 The complex zeros of f are ? glamrockqueen7 2021-09-07 Answered
Find the complex zeros of the following polynomial function. Write f in factored form.
${x}^{4}+2{x}^{3}+22{x}^{2}+50x-75$
The complex zeros of f are ?
Use the complex zeros to factor f.
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Step 1
Given function is,
$f\left(x\right)={x}^{4}+2{x}^{3}+22{x}^{2}+50x-75$
The objective is to find the complex zeros of the function also write the function in factored form.
Step 2
Consider the function,
$f\left(x\right)={x}^{4}+2{x}^{3}+22{x}^{2}+50x-75$
Since, $x=1$ satisfies the function.
Hence, $\left(x-1\right)$ is a factor of the function.
Then,
$f\left(x\right)={x}^{4}+2{x}^{3}+22{x}^{2}+50x-75$
$={x}^{3}\left(x-1\right)+3{x}^{2}\left(x-1\right)+25x\left(x-1\right)+75\left(x-1\right)$
$=\left(x-1\right)\left({x}^{3}+3{x}^{2}+25x+75\right)$
Now, $x=-3$ also satisfy the function.
Hence, the $\left(x+3\right)$ is a factor of the function.
Then function can be written as,
$f\left(x\right)=\left(x-1\right)\left({x}^{3}+3{x}^{2}+25x+75\right)$
$=\left(x-1\right)\left({x}^{2}\left(x+3\right)+25\left(x+3\right)\right)$
$=\left(x-1\right)\left(x+3\right)\left({x}^{2}+25\right)$
Now, $x=-5i\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}5i$ satisfy the function.
Hence , function can be written as,
$f\left(x\right)=\left(x-1\right)\left(x+3\right)\left(x-5i\right)\left(x+5i\right)$
Therefore, the complex zeros of the function are $x=-5i\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}5i$
Step 3
The function in factored form: $f\left(x\right)=\left(x-1\right)\left(x+3\right)\left(x-5i\right)\left(x+5i\right)$