Using the Factor Theorem, the polynomial is given by: [tex]f(x) = (x + 4)^4(x + 1)^3(x - 6)^6[/tex]
- The graph is sketched at the end of the answer.
The Factor Theorem states that a polynomial function with roots [tex]x_1, x_2, \cdots, x_n[/tex] is given by:
[tex]f(x) = a(x - x_1)(x - x_2) \cdots (x - x_n)[/tex]
In which a is the leading coefficient.
In this problem, the roots are:
- Root of -4 with multiplicity 4, hence [tex]x_1 = x_2 = x_3 + x_4 = -4[/tex].
- Root of -1 with multiplicity 3, hence [tex]x_5 = x_6 = x_7 = 3[/tex].
- Root of 5 with multiplicity 6, hence [tex]x_8 = x_9 = x_10 = x_11 = x_12 = x_13 = 6[/tex]
Then:
[tex]f(x) = a(x - (-4))^4(x - (-1))^3(x-6)^6[/tex]
[tex]f(x) = a(x + 4)^4(x + 1)^3(x - 6)^6[/tex]
- Positive leading coefficient, hence [tex]a = 1[/tex].
- 13th degree, so it is odd.
Then:
[tex]f(x) = (x + 4)^4(x + 1)^3(x - 6)^6[/tex]
At the end of the answer, an sketch of the graph is given.
For more on the Factor Theorem, you can check https://brainly.com/question/24380382
