Question

Complete the statements about the key features of the graph of f(x) = x5 – 9x3. As x goes to negative infinity, f(x) goes to negative infinity, and as x goes to positive infinity, f(x) goes to positive infinity.
Choose all of the zeroes of f(x).
–3 with multiplicity 1
3 with multiplicity 1
0 with multiplicity 1
0 with multiplicity 3
3 with multiplicity 0

Answer 1

The function is $f(x)= x^{5} -9x ^{3}$

1. let's factorize the expression $x^{5} -9x ^{3}$:

$f(x)= x^{5} -9x ^{3}= x^{3} ( x^{2} -9)=x^{3}(x-3)(x+3)$

the zeros of f(x) are the values of x which make f(x) = 0.

from the factorized form of the function, we see that the roots are:

-3, multiplicity 1
3, multiplicity 1
0, multiplicity 3

(the multiplicity of the roots is the power of each factor of f(x) )


2.  
The end behavior of f(x), whose term of largest degree is $x^{5}$, is the same as the end behavior of  $x^{3}$, which has a well known graph. Check the picture attached. 

(similarly the end behavior of an even degree polynomial, could be compared to the end behavior of $x^{2}$)

so, like the graph of $x^{3}$, the graph of $f(x)= x^{5} -9x ^{3}$ :

"As x goes to negative infinity, f(x) goes to negative infinity, and as x goes to positive infinity, f(x) goes to positive infinity. "

Answer 2

The end behavior of f(x) has the same end behavior of $x^3$ and all the zeroes of f(x) wiil be: -3 with multiplicity 1, 3 with multiplicity 1, 0 with multiplicity 3.

Given :

$f(x) = x^5-9x^3$   --- (1)

The value of x at which f(x) becomes zero can be determine by eqaution the equation (1) to zero.

$x^5-9x^3=0$

$x^3(x^2-9)=0$

$x^3(x-3)(x+3)=0$

Therefore, all the zeroes of f(x) will be:

-3 with multiplicity 1,

3 with multiplicity 1,

0 with multiplicity 3.

The end behavior of f(x) has the same end behavior of $x^3$. As x goes to negative infinity, f(x) goes to negative infinity, and as x goes to positive infinity, f(x) goes to positive infinity.The graph of f(x) is attached below.

For more information, refer the link given below:

https://brainly.com/question/24153248