Question

Analyze each polynomial function.

$f\left(x\right)=-x^5-x^4+x^3+x^2$

Short answer

The given function is:

$f\left(x\right)=-x^5-x^4+x^3+x^2$

Step-by-step solution

Given Information

The given function is:

$f\left(x\right)=-x^5-x^4+x^3+x^2$

Find the degree of the polynomial

Notice that, We have a degree of 5 from $f\left(x\right)=-x^5-x^4+x^3+x^2$ Thus, The graph of $f$ has an end behaviour similar to$y=-x^5$

Find the x intercept

Now let's determine the intercepts of $f$To get the $x$-intercept, equate $f\left(x\right)$to zero

factor out $-x^2$

$-x^5-x^4+x^3+x^2=0$

factor by grouping

$-x^2\left(x^3+x^2-x-1\right)=0$

$-x^2(x+1{)}^2(x-1)=0$

$$\begin{gathered} -x^2=0 \\ x=0 \end{gathered}$$

or

$$\begin{gathered} x+1=0 \\ x=-1 \end{gathered}$$

or

$$\begin{gathered} x-1=0 \\ x=1 \end{gathered}$$

So, the $x$-intercepts are $x=0,x=-1,x=1$Considering $f\left(0\right)=0,$thus the $y$-intercept is $(0,0)$

• Note that the zeros of $f$are: 0 with the multiplicity of $2$is a zero that has an even multiplicity,
• Thus the graph touches the $x$-axis, $-1$with the multiplicity of $2$is a zero that has an even multiplicity, Thus the graph touches the $x$-axis
• Also, 1 with the multiplicity of 1 is a zero that has an odd multiplicity, Thus the graph crosses the x-axis

From the graph above, we can also see that the MAXIMUM turning point of the function is at point $(-0.54,0.095)$and $(0.74,0.4311)$and the MINIMUM turning point is at point $(-1,0)$and $(0,0)$

• Therefore, the function has a domain and range of all real numbers; or simply $(-\infty ,\infty )$
• By observing the graph above We say that the function At intervals of $(-1,-0.54)$and $(0,0.74)$is increasing; At intervals of $(-\infty ,-1),(-0.54,0)$and $(0.74,\infty )$is decreasing

The graph