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Answer:
Part A: Yes.
Part B: y approaches negative infinity.
Step-by-step explanation:
We're told to determine if the function has zeroes or solutions of x = -5, -3, 1.
A common way to determine the solutions of a polynomial is to factor. But in this case, it's not optimal; while the second and third terms can be reduced --to -7x(x+1)--, the remaining terms cannot do the same.
So, we rely on the Factor theorem. It states that if p is a solution to a function, then f(p) = 0.
Checking the Zeroes:
x = -5:
[tex]-(-5)^3-7(-5)^2-7(-5)+15[/tex]
[tex]=125-175+35+15[/tex]
[tex]=0[/tex].
x = -3:
[tex]-(-3)^3-7(-3)^2-7(-3)+15[/tex]
[tex]=27-63+21+15[/tex]
[tex]=0[/tex].
x = 1:
[tex]-(1)^3-7(1)^2-7(1)+15[/tex]
[tex]-1-7-7+15[/tex]
[tex]=0[/tex].
Thus, they're all solutions to f(x).
[tex]\hrulefill[/tex]
The leading coefficient of a cubic function determines the start and end behavior.
A negative leading coefficient indicates a start behavior where y approaches positive infinity and an end behavior where the y value approaches negative infinity.
Putting Numbers to the Conclusion
Plugging in x = 3 into -x³ is -27, and plugging a bigger number of 100 is -1,000,000.
And since the cubed term is always the biggest value, the overall value of the function will also have a value close to that term.
Paying attention to our plugged values, as x gets bigger (in the positive direction), the y-value gets bigger in the negative direction.
From this, we can accurately hypothesize that the end behavior (as x approaches positive infinity) will be in the negative infinity direction.