Write a cubic function that passes through the points. (-3, 0) (-1, 10) (0,0) (4, 0) Please explain your work, thank you! :)

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Let f(x) be the cubic function.

Then, we will write the function into the standard format:

f(x) = ax^3 + bx^2 + cx + d

Now to dteremine a,b,c,and d values, we will subsitute the points given.

The point (0,0):

==> f(0)=0+0++0+d= 0

==> d= 0

The point (4,0)

==>...

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Let f(x) be the cubic function.

Then, we will write the function into the standard format:

f(x) = ax^3 + bx^2 + cx + d

Now to dteremine a,b,c,and d values, we will subsitute the points given.

The point (0,0):

==> f(0)=0+0++0+d= 0

==> d= 0

The point (4,0)

==> f(4) = a*4^3 + b*4^2 + c*4 = 0

==> 64a + 16b + 4c = 0

Divide by 4 to simplify:

==> 16a + 4b + c = 0.............(1)

Now the point (-3,0)

f(-3) = a*-3^3 + b*-3^2 + c*-3 = 0

==> -27a + 9b -3c = 0

divide by 3:

==> -9a + 3b -c = 0 ..............(2)

Now the point (-1,10)

f(-1) = a*-1^3 + b*-1^2 + c*-1 = 10

==> -a + b -c = 10...................(3)

Now we will solve the system:

using the elimination method, we will add (1) and (2):

==> 7a +7b = 0

divide by 7:

==> a+ b = 0...............(4)

Now add (1) and (3):

==> 15a +5b = 10

Divide by 5 :

==> 3a + b = 2.............(5)

Now subtract (4) from (5)

==> 2a = 2

==> a= 1

==> b= -1

==> c= -12

==> f(x) = x^3 - x^2 -12x

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