f(x)= x4 + x3 – 13x2 – 1x + 12 f(x)=x^4 + x^3 - 13x^2 - 1x+12 graph the following polynomial graphs on graph paper. You will...

We'll determine x and y
intercepts.

The graph is intercepting x axis for y =
0.

Since y = f(x) => x^4 + x^3 - 13x^2 - x + 12 =
0

Since all the coefficients of the given function are
integer numbers, the roots of the function can be found among the divisors of the free
coefficient, 12.

We'll check if x = -4 is one of the
roots.

f(-4) = 256 - 64 - 208 + 4 +
12

f(-4) = 0 => x = -4 is one of the roots of the
function.

Applying long division, we'll get the quotient q
= x^3 - 3x^2 - x + 3.

We'll apply the reminder
theorem:

(x + 4)(x^3 - 3x^2 - x + 3) =
f(x)

If f(x) = 0 => (x + 4)(x^3 - 3x^2 - x +
3)

(x + 4)[x^2*(x - 3) - (x - 3)] =
0

(x + 4)(x - 3)(x^2 - 1) =
0

The difference of squares returns the
product:

x^2 - 1 = (x - 1)(x +
1)

(x + 4)(x - 3)(x - 1)(x + 1) =
0

We'll cancel each factor:

x
+ 4 = 0 => x = -4

x - 3 = 0 => x =
3

x - 1 = 0 => x = 1

x
+ 1 = 0 => x = -1

The graph is intercepting x axis
in the points: (4, 0), (3, 0), (-1, 0), (1, 0).

The graph 
is intercepting y axis if x = 0.

f(0) = 0^4 + 0^3 – 13*0^2
– 0 + 12

f(0) = 12

The graph 
is intercepting y axis in the point (0, 12).

The end
behavior of the function is found when is evaluated the limit of the function if x is
approaching to +/-infinite:

x approaches to +
infinite:

lim x^4 + x^3 - 13x^2 - x+12 = (+infinite)^4 =
+infinite

x approaches to -
infinite:

lim x^4 + x^3 - 13x^2 - x+12 = (-infinite)^4 =
+infinite