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A 5th degree polynomial with a root of multiplicity 2 at x = 1, and a root of multiplicity 3 at x = -4, can be written as the function y=(x-1)^a(x-b)^c where​:

a =

b =

c =

The y-intercept of this polynomial is (0, ).


NOTE: Your answers should be integers.

Respuesta :

Using the Factor Theorem, it is found that:

  • The coefficients are: a = 2, b = -4, c = 3.
  • The y-intercept of this polynomial is (0, 64).

What is the Factor Theorem?

  • The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \codts, x_n[/tex] is given by:

[tex]f(x) = (x - x_1)(x - x_2) \cdots (x - x_n)[/tex]

In this problem:

  • Root of multiplicity 2 at x = 1, hence [tex]x_1 = x_2 = 1[/tex].
  • Root of multiplicity 3 at x = -4, hence [tex]x_3 = x_4 = x_5 = -4[/tex]

Then:

[tex]f(x) = (x - x_1)(x - x_2)(x - x_3)(x - x_4)(x - x_5)[/tex]

[tex]f(x) = (x - 1)(x - 1)(x + 4)(x + 4)(x + 4)[/tex]

[tex]f(x) = (x - 1)^2(x + 4)^3[/tex]

Hence the coefficients are: a = 2, b = -4, c = 3.

For the y-intercept, we have that:

[tex]f(0) = (0 - 1)^2(x + 4)^3 = 4^3 = 64[/tex]

The y-intercept of this polynomial is (0, 64).

To learn more about the Factor Theorem, you can take a look at https://brainly.com/question/24380382

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