Answer: We can plot the graph with help of below explanation.
Step-by-step explanation:
Since, given equation of polynomial,
![P(x) = x^4 - 3x^3 - 8x^2 + 12x + 16](https://tex.z-dn.net/?f=P%28x%29%20%3D%20x%5E4%20-%203x%5E3%20-%208x%5E2%20%2B%2012x%20%2B%2016)
End behavior : Since, the leading coefficient of the polynomial is positive and even.
Therefore, the end behavior of the polynomial is,
as ![x\rightarrow -\infty](https://tex.z-dn.net/?f=x%5Crightarrow%20-%5Cinfty)
And,
as ![x\rightarrow +\infty](https://tex.z-dn.net/?f=x%5Crightarrow%20%2B%5Cinfty)
Points of the curve : since, P(4) = 0
Therefore, (x-4) is the multiple of P(x),
And we can write, ![x^4 - 3x^3 - 8x^2 + 12x + 16= (x-4)(x^3+x^2-4x-4)](https://tex.z-dn.net/?f=x%5E4%20-%203x%5E3%20-%208x%5E2%20%2B%2012x%20%2B%2016%3D%20%28x-4%29%28x%5E3%2Bx%5E2-4x-4%29)
![x^4 - 3x^3 - 8x^2 + 12x + 16=(x-4)(x+1)(x^2-4)](https://tex.z-dn.net/?f=x%5E4%20-%203x%5E3%20-%208x%5E2%20%2B%2012x%20%2B%2016%3D%28x-4%29%28x%2B1%29%28x%5E2-4%29)
![x^4 - 3x^3 - 8x^2 + 12x + 16= (x-4)(x+1)(x+2)(x-2)](https://tex.z-dn.net/?f=x%5E4%20-%203x%5E3%20-%208x%5E2%20%2B%2012x%20%2B%2016%3D%20%28x-4%29%28x%2B1%29%28x%2B2%29%28x-2%29)
Thus, the roots of equation are 4, 2, -1 and -2.
Therefore, x-intercepts of the polynomial are (4,0) (2,0) (-1,0) and (-2,0)
Also, the y-intercept of the polynomial is ( 0,16)
Thus, we can plot the graph with help of the above information.