Respuesta :

kudzordzifrancis

Answer:

a=-5

b=4

c=-1

d=0

Step-by-step explanation:

The given cubic equation is of the form;

[tex]y = a {x}^{3} + b {x}^{2} + cx + d[/tex]

When x=0 and y=0, we get:

[tex]0= a ({0})^{3} + b( {0})^{2} + c(0) + d[/tex]

[tex]d = 0[/tex]

when x=1 and y=-2, we get:

[tex] - 2= a ({1})^{3} + b( {1})^{2} + c(1) \\ a + b + c = - 2[/tex]

When x=2 and y=-26, we get:

[tex] - 26= a ({2})^{3} + b( {2})^{2} + c(2) + d \\ - 26 = 8a + 4b + 2c \\ 4a + 2b + c = - 13[/tex]

When x=3 and y=-54, we get:

[tex] - 54= a ({3})^{3} + b( {3})^{2} + c(3) + d \\ - 54 = 27a + 9b + 3c \\ 9a + 3b + c = - 18[/tex]

We need to solve the system:

[tex]a + b + c = - 2 \\ 4a + 2b + c = - 13 \\ 9a + 3b + c = - 18[/tex]

From the top equation:

[tex]c = - 2 - a - b[/tex]

Put this expression into equation 2 and 3

[tex]4a + 2b - 2 - a - b = - 13 \\ 3a + b = - 11[/tex]

also

[tex]9a + 3b - 2 - a - b = - 18 \\ 8a + 2b = - 16 \\ 4a + b = - 8[/tex]

Now we solve the system:

[tex]3a + b = - 11 \\ 4a + b = - 16[/tex]

subtract the top equation from the down one to get:

[tex]4a - 3a = - 16 - - 11 \\ a = - 5[/tex]

This implies that:

[tex]3( - 5) + b = - 11 \\ - 15 + b = - 11 \\ b = - 11 + 15 = 4[/tex]

and

[tex]c = - 2 - - 5 - 4 = - 1[/tex]

MrRoyal

A cubic function has a leading exponent of 3

The values of the coefficients are: [tex]a = 3[/tex], [tex]b=-20[/tex], [tex]c = 15[/tex] and [tex]d = 0[/tex]

The cubic function is given as:

[tex]y = ax^3 + bx^2 + cx + d[/tex]

When x = 0, y = 0.

So, we have:

[tex]a(0)^3 + b(0)^2 + c(0) + d = 0[/tex]

[tex]d = 0[/tex]

When x = 1, y = -2.

So, we have:

[tex]a(1)^3 + b(1)^2 + c(1) + d = -2[/tex]

[tex]a + b + c + d = -2[/tex]

Substitute 0 for d

[tex]a + b + c = -2[/tex]

When x = 2, y = -26.

So, we have:

[tex]a(2)^3 + b(2)^2 + c(2) = -26[/tex]

[tex]8a + 4b + 2c = -26[/tex]

Divide through by 2

[tex]4a + 2b + c = -13[/tex]

When x = 3, y = -54.

So, we have:

[tex]a(3)^3 + b(3)^2 + c(3) = -54[/tex]

[tex]27a + 9b + 3c= -54[/tex]

Divide through by 3

[tex]9a + 3b + c= -18[/tex]

So, we have:

[tex]a + b + c = -2[/tex]

[tex]4a + 2b + c = -13[/tex]

[tex]9a + 3b + c= -18[/tex]

Make c the subject

[tex]c = -2 -a -b[/tex]

[tex]c = -13 -4a -2b[/tex]

[tex]c = -18 -9a -3b[/tex]

Equate all the three equations

[tex]-2-a-b = -13-4a-2b[/tex]

[tex]-2-a-b = -18 - 9a -3b[/tex]

Simplify the equations

[tex]4a -a +2b - b = 2 - 13[/tex]

[tex]3a +b = -11[/tex]

[tex]9a -a+3b -b =2-18[/tex]

[tex]8a+2b =-16[/tex]

Divide through by 2

[tex]4a+b =-8[/tex]

Subtract [tex]4a+b =-8[/tex] and [tex]3a +b = -11[/tex]

[tex]4a - 3a + b - b =-8 + 11[/tex]

[tex]a =3[/tex]

Substitute 3 for a in [tex]4a+b =-8[/tex]

[tex]4(3) + b = -8[/tex]

[tex]12 + b = -8[/tex]

Subtract 12 from both sides

[tex]b = -20[/tex]

Recall that:

[tex]c = -2 -a -b[/tex]

So, we have:

[tex]c = -2 -3 + 20[/tex]

[tex]c = 15[/tex]

Hence, the values of the coefficients are:

[tex]a = 3[/tex]

[tex]b=-20[/tex]

[tex]c = 15[/tex]

[tex]d = 0[/tex]

Read more about cubic functions at:

https://brainly.com/question/20896994

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