Answer:
3 solutions
Step-by-step explanation:
You want to know the number of solutions (zeros) of the cubic function ...
f(x) = -x³ -32x² +12x +40
Cubic function
Any 3rd-degree polynomial function has 3 zeros. (This is the fundamental theorem of algebra.)
Signs of roots
Descartes' rule of signs says the one sign change in the coefficients (--++) means there is one positive real root. On its face, this means there will be 0 or 2 negative real roots.
Values of roots
The value of f(0) is positive (40), and the value of f(-1) is negative (-3), so there must be two negative real roots. (F(x) is positive for very large negative x, so there must be two x-intercepts for x < 0.)
A graph shows the roots to be irrational. The three real roots are approximately ...
- −32.3328769399
- −0.958209309898
- 1.29108624985
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Additional comment
The "steps" are to find the degree of the polynomial. It is 3, so there are 3 zeros. It is not complicated.
What is more complicated is finding their values. The checks we described above can tell whether there are 1 or 3 real roots. (An odd-degree polynomial always has at least 1 real root.) A graph is helpful for finding whether the roots are real or complex, rational or irrational.
Here, any rational roots must be integer divisors of the constant term, 40. We know the root between 0 and -1 is irrational, so all of them are.
The above root values were obtained by Newton's Method iteration, starting from the approximate values shown on the graph.
