<em>Note: Since you missed to mention the the expression of the function </em>
<em> . After a little research, I was able to find the complete question. So, I am assuming the expression as </em>
<em> and will solve the question based on this assumption expression of </em>
<em>, which anyways would solve your query.</em>
Answer:
As
![p\left(-2\right)=0](https://tex.z-dn.net/?f=p%5Cleft%28-2%5Cright%29%3D0)
Therefore,
is a root of the polynomial <em> </em>![p(x)=x^4-9x^2-4x+12](https://tex.z-dn.net/?f=p%28x%29%3Dx%5E4-9x%5E2-4x%2B12)
As
![p\left(2\right)=-16](https://tex.z-dn.net/?f=p%5Cleft%282%5Cright%29%3D-16)
Therefore,
is not a root of the polynomial <em> </em>
Step-by-step explanation:
As we know that for any polynomial let say<em> </em>
<em>, </em>
is the root of the polynomial if
.
In order to find which of the given values will be a root of the polynomial,
<em>, </em>we must have to evaluate <em> </em>
<em> </em>for each of these values to determine if the output of the function gets zero.
So,
Solving for ![p\left(-2\right)](https://tex.z-dn.net/?f=p%5Cleft%28-2%5Cright%29)
<em> </em>![p(x)=x^4-9x^2-4x+12](https://tex.z-dn.net/?f=p%28x%29%3Dx%5E4-9x%5E2-4x%2B12)
![p\left(-2\right)=\left(-2\right)^4-9\left(-2\right)^2-4\left(-2\right)+12](https://tex.z-dn.net/?f=p%5Cleft%28-2%5Cright%29%3D%5Cleft%28-2%5Cright%29%5E4-9%5Cleft%28-2%5Cright%29%5E2-4%5Cleft%28-2%5Cright%29%2B12)
![\mathrm{Simplify\:}\left(-2\right)^4-9\left(-2\right)^2-4\left(-2\right)+12:\quad 0](https://tex.z-dn.net/?f=%5Cmathrm%7BSimplify%5C%3A%7D%5Cleft%28-2%5Cright%29%5E4-9%5Cleft%28-2%5Cright%29%5E2-4%5Cleft%28-2%5Cright%29%2B12%3A%5Cquad%200)
![\left(-2\right)^4-9\left(-2\right)^2-4\left(-2\right)+12](https://tex.z-dn.net/?f=%5Cleft%28-2%5Cright%29%5E4-9%5Cleft%28-2%5Cright%29%5E2-4%5Cleft%28-2%5Cright%29%2B12)
![\mathrm{Apply\:rule}\:-\left(-a\right)=a](https://tex.z-dn.net/?f=%5Cmathrm%7BApply%5C%3Arule%7D%5C%3A-%5Cleft%28-a%5Cright%29%3Da)
![=\left(-2\right)^4-9\left(-2\right)^2+4\cdot \:2+12](https://tex.z-dn.net/?f=%3D%5Cleft%28-2%5Cright%29%5E4-9%5Cleft%28-2%5Cright%29%5E2%2B4%5Ccdot%20%5C%3A2%2B12)
![\mathrm{Apply\:exponent\:rule}:\quad \left(-a\right)^n=a^n,\:\mathrm{if\:}n\mathrm{\:is\:even}](https://tex.z-dn.net/?f=%5Cmathrm%7BApply%5C%3Aexponent%5C%3Arule%7D%3A%5Cquad%20%5Cleft%28-a%5Cright%29%5En%3Da%5En%2C%5C%3A%5Cmathrm%7Bif%5C%3A%7Dn%5Cmathrm%7B%5C%3Ais%5C%3Aeven%7D)
![=2^4-2^2\cdot \:9+8+12](https://tex.z-dn.net/?f=%3D2%5E4-2%5E2%5Ccdot%20%5C%3A9%2B8%2B12)
![=2^4+20-2^2\cdot \:9](https://tex.z-dn.net/?f=%3D2%5E4%2B20-2%5E2%5Ccdot%20%5C%3A9)
![=16+20-36](https://tex.z-dn.net/?f=%3D16%2B20-36)
![=0](https://tex.z-dn.net/?f=%3D0)
Thus,
![p\left(-2\right)=0](https://tex.z-dn.net/?f=p%5Cleft%28-2%5Cright%29%3D0)
Therefore,
is a root of the polynomial <em> </em>
<em>.</em>
Now, solving for ![p\left(2\right)](https://tex.z-dn.net/?f=p%5Cleft%282%5Cright%29)
<em> </em>![p(x)=x^4-9x^2-4x+12](https://tex.z-dn.net/?f=p%28x%29%3Dx%5E4-9x%5E2-4x%2B12)
![p\left(2\right)=\left(2\right)^4-9\left(2\right)^2-4\left(2\right)+12](https://tex.z-dn.net/?f=p%5Cleft%282%5Cright%29%3D%5Cleft%282%5Cright%29%5E4-9%5Cleft%282%5Cright%29%5E2-4%5Cleft%282%5Cright%29%2B12)
![\mathrm{Remove\:parentheses}:\quad \left(a\right)=a](https://tex.z-dn.net/?f=%5Cmathrm%7BRemove%5C%3Aparentheses%7D%3A%5Cquad%20%5Cleft%28a%5Cright%29%3Da)
![p\left(2\right)=2^4-9\cdot \:2^2-4\cdot \:2+12](https://tex.z-dn.net/?f=p%5Cleft%282%5Cright%29%3D2%5E4-9%5Ccdot%20%5C%3A2%5E2-4%5Ccdot%20%5C%3A2%2B12)
![p\left(2\right)=2^4-2^2\cdot \:9-8+12](https://tex.z-dn.net/?f=p%5Cleft%282%5Cright%29%3D2%5E4-2%5E2%5Ccdot%20%5C%3A9-8%2B12)
![p\left(2\right)=2^4+4-2^2\cdot \:9](https://tex.z-dn.net/?f=p%5Cleft%282%5Cright%29%3D2%5E4%2B4-2%5E2%5Ccdot%20%5C%3A9)
![p\left(2\right)=16+4-36](https://tex.z-dn.net/?f=p%5Cleft%282%5Cright%29%3D16%2B4-36)
![p\left(2\right)=-16](https://tex.z-dn.net/?f=p%5Cleft%282%5Cright%29%3D-16)
Thus,
![p\left(2\right)=-16](https://tex.z-dn.net/?f=p%5Cleft%282%5Cright%29%3D-16)
Therefore,
is not a root of the polynomial <em> </em>
<em>.</em>
Keywords: polynomial, root
Learn more about polynomial and root from brainly.com/question/8777476
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