Question

Analyze each polynomial function using Steps $1$through $8$:

$f\left(x\right)=x^4-x^3-6x^2+4x+8$.

Short answer

The zeroes of the polynomial are $-2,-1,$and $2$.

The graph for the function is,

Step-by-step solution

Step 1. Determining the end behavior of the graph of the function. 

The polynomial function $f$is of degree$4$. The graph behaves like $y=x^4$for large values of$\left|x\right|$.

Step 2. Finding the x and y intercepts.

Setting $y=0$for the equation

$f\left(x\right)=x^4-x^3-6x^2+4x+8$,

The $y-$intercept is $(0,8)$.

The$x-$intercept is$(-2,0),(-1,0),$and$(2,0)$.

Step 3. Determine the zeros of the function and their multiplicity. 

The zeros of $f$are $-2,-1,$and $2$.

The zero $-2$is a zero of multiplicity $1$, so the graph of $f$crosses the $x-$axis at $x=-2$.

The zero $-1$is a zero of multiplicity $1$, so the graph of $f$crosses the $x-$axis at $x=-1$.

The zero$2$is a zero of multiplicity$2$, so the graph of$f$crosses the$x-$axis at$x=2$.

Step 4. Graph the function.

The graph for the function,

$f\left(x\right)=x^4-x^3-6x^2+4x+8$is,

Step 5. Approximate the turning points of the graph.

The turning points of the polynomial function are,

$(-\frac{5}{8}-\frac{1}{8}\sqrt{57},-3.12),(-\frac{5}{8}+\frac{1}{8}\sqrt{57},8.64),$and$(2,0)$.

Step 6. Using the information draw the graph.

The graph of the function by hand is,

Step 7. Finding the domain and range of the function.

The domain of the function is $\mathrm{ℝ}$which is $(-\infty ,\infty )$.

The range of the function is$(-1.6,-3.11)$.

Step 8. Where the function is increasing and where it is decreasing.

The polynomial function increases from $(-\frac{5}{8}-\frac{1}{8}\sqrt{57},-\frac{5}{8}+\frac{1}{8}\sqrt{57})\cup (2,\infty )$.

The polynomial function decreases from,

$(-\infty ,-\frac{5}{8}-\frac{1}{8}\sqrt{57})\cup (-\frac{5}{8}+\frac{1}{8}\sqrt{57},2)$.