Answer: We can plot the graph with help of below explanation.
Step-by-step explanation:
Since, given equation of polynomial,
[tex]P(x) = x^4 - 3x^3 - 8x^2 + 12x + 16[/tex]
End behavior : Since, the leading coefficient of the polynomial is positive and even.
Therefore, the end behavior of the polynomial is,
[tex]f(x)\rightarrow -\infty[/tex] as [tex]x\rightarrow -\infty[/tex]
And, [tex]f(x)\rightarrow +\infty[/tex] as [tex]x\rightarrow +\infty[/tex]
Points of the curve : since, P(4) = 0
Therefore, (x-4) is the multiple of P(x),
And we can write, [tex]x^4 - 3x^3 - 8x^2 + 12x + 16= (x-4)(x^3+x^2-4x-4)[/tex]
[tex]x^4 - 3x^3 - 8x^2 + 12x + 16=(x-4)(x+1)(x^2-4)[/tex]
[tex]x^4 - 3x^3 - 8x^2 + 12x + 16= (x-4)(x+1)(x+2)(x-2)[/tex]
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.
