JonathanFlorez479
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Suppose the graph of a cubic polynomial function has the same zeroes and passes through the coordinate (0, -5). Write the equation of this cubic polynomial function. Recall that the zeros are (2, 0), (3, 0), and (5, 0). What is the y-intercept of this graph?

Respuesta :

In this we know all three zeros and one point from which the graph pass.
So we will let specific cubic polynomial function of the form
[tex]f(x) = a(x - x_1)(x-x_2)(x-x_3)[/tex]

As we know zeros are that point where we will get value of function equal to zero. So it is basically in form [tex](x_n , 0)[/tex]

SO in given question zeros are (2 , 0) , (3, 0) and (5,0)
So we can say [tex]x_1 = 2 , x_2 = 3 , x_3 = 5[/tex]

So required equation is
[tex]f(x) = a (x-2)(x-3)(x-5)[/tex]
              [tex]= a[(x^2 - 2x - 3x + 6)(x-5)][/tex]
              [tex]= a[(x^2 - 5x+6)(x-5)][/tex]
              [tex]= a(x^3 - 5x^2 + 6x- 5x^2 + 25x - 30)[/tex]
              [tex]= a(x^3 - 10x^2+31x-30)[/tex]
Now we have one point (0 , -5) from which graph passes.
So we say at x = 0 , f(x) = -5
[tex]-5= a (0-0+0 - 30)[/tex]
[tex]-5 = -30a[/tex]
[tex]a = \frac{-5}{-30} = \frac{1}{6} [/tex]
So required equation of cubic polynomial is
[tex]f(x) = \frac{1}{6}(x^3 -10x^2+31x-30) [/tex]

For finding y - intercept we simply plugin x = 0 in given equation.
As we know at x = 0 , value of function is -5.
So y - intercept is -5.
MrRoyal

The equation of the function is [tex]y = \frac 16(x -2)(x -3)(x -5)[/tex], and the y-intercept of the function is -5

A cubic function is represented as:

[tex]y = a(x -x_1)(x -x_2)(x -x_3)[/tex]

The zeros are (2, 0), (3, 0), and (5, 0).

This means that:

(x1, y) = (2, 0)

(x2, y) = (3, 0)

(x3,y) = (5, 0)

So, we have:

[tex]y = a(x -2)(x -3)(x -5)[/tex]

The graph passes through the point (0,-5).

So, we have:

[tex]-5 = a(0 -2)(0 -3)(0 -5)[/tex]

Evaluate the products

[tex]-5 = -30a[/tex]

Solve for a

[tex]a = \frac{5}{30}[/tex]

[tex]a = \frac{1}{6}[/tex]

Substitute 1/6 for a in [tex]y = a(x -2)(x -3)(x -5)[/tex]

[tex]y = \frac 16(x -2)(x -3)(x -5)[/tex]

The y-intercept of the function is when x = 0.

Hence, the y-intercept of the function is -5

Read more about cubic functions at:

https://brainly.com/question/20896994

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