Question

The following polynomial represents a profit function for a certain production line,
where x is a number of produced units,
find
a. the zeros and the multiplicity of each
b. where the graph crosses or touches the x-axis
c. number of turning points
d. explain the meaning of the above, if any
f(x)=(x+5)(x−3)(x+3)

Answer 1

graph crosses the x axis at y = 0
y= 0 when either x+5 = 0
x-3 = 0
and/or x+3 = 0
(as anything times 0 = 0)
so from that you can get your turning points:
x+5 = 0, therefore x=-5
x-3 = 0, therefore x=3
x+3 = 0, therefore x=-3

your function crosses the x axis at the points x= -5, -3, and 3

Answer 2

you following ok so far, do you want anything explained a bit more?

Answer 3

thank you its just so confusing math hates me lol

Answer 4

sók, do you want me to move on to the next part or do you not understand the above yet and want a bit more detail...?

Answer 5

no go ahead i will have to go over this about 5 times to understand it but i will get it ;)

Answer 6

ok, this is what it looks like:

Answer 7

so as you can see, it's a cubic polynomial, with 2 turning points
to work them out mathematically, multiply out the brackets so you're left with the form:
f(x) = ax^3 + bx^2 + cx +d

in this case:
f(x) = (x+5)(x−3)(x+3)
f(x) = x^3 + 5x^2 -9x -45

Answer 8

now to find the turning points:
they will occur when the gradient of the graph = 0
the equation for gradient is the derivative of the equation of the function

so, the equation for your function:

f(x) = x^3 + 5x^2 -9x -45

therefore derivative of function:

f '(x) = 3x^2 + 10x -9

Answer 9

now you can use the quadratic formula to find the exact position of when

3x^2 + 10x -9 = 0, which is the x co-ordinates of the turning points

in this case it's at:
x= -4.07 ish and
x= 0.737 ish

Answer 10

so as it's a profitability graph, and x = number of units produced... i'd assume that y = profits made...? maybe...?
i'm not sure how to do the explanation part, sorry, just the maths
and i don't know what a zero or a multiplicity is...?
do you have any ideas?

Answer 11

multiplicity i believe it is what you multiply like if it had^2 in it it would be 2. or if it had ^3 answer would be 3.as for the zero i dont know