Which is the graph of the function f(x) = x3 + 6x2 + 11x + 6?

f(x) = x^3 + 6x2 + 11x + 6 begins with a positive 3rd power of x. Immediately we know that the graph begins in Quadrant 3 and continues (not ends) in Quadrant 1. Looking at the last term, we see that possible factors may be 6, 3, 2 or 1 or one or more of the negatives. Supposing I chose -2 and test whether it is or is not a root of this function:

_____________
-2 / 1 6 11 6
-2 -8 -6
----------------------
1 4 3 0

Because the remainder here is zero, we know that -2 is a root. Thus, the graph passes thru (-2,0). From the coefficients 1 4 3, we see that -1 and -3 are the other roots. The graph crosses the x-axis at x = -1, x= -3 and x= -2.
Can you now identify the graph that properly represents f(x) = x3 + 6x2 + 11x + 6?

psm22415 psm22415 · Answer 2 · 2022-01-14T08:46:42+01:00

The graph of the function is attached below.

The roots of the given equation are -3, -2, and -1.

We have to determine the graph of the function f(x) = x3 + 6x2 + 11x + 6.

According to the question,

The graph of the function is determined by factorizing the function following all the steps given below.

[tex]\rm f(x) = x^3 + 6x^2 + 11x + 6.[/tex]

Factorize the equation,

[tex]\rm x^3 +6x^2+ 11x + 6\\\\ x^3+3x^2+3x^2 + 9x+2x+ 6\\\\ x(x^2+3x+2) +3(x^2+3x+2)\\\\(x+3) (x^2+3x+2)\\\\(x+3) (x+1) (x+2)[/tex]

The roots of the equation are as follows;

[tex]\rm x+3=0, \ \ x=-3\\\\x+2=0,\ \ x=-2\\\\x+1=0, \ \ x=-1[/tex]

Hence, the roots of the given equation are -3, -2, and -1.

To know more about the Cubic equation following all the steps given below.

https://brainly.com/question/24201315