Respuesta :
The polynomial is:
[tex]p(x) = x^4 -4x^3-3x^2 +10x+8[/tex]
Solution:
Given that,
A polynomial, p(x), has a lead coefficient of 1 and exactly three distinct zeros
The zeros are:
x = -1 is a zero of multiplicity two
x = 2 is a zero of multiplicity one
x = 4 is a zero of multiplicity one
Zero of multiplicity means "How many times a particular number is a zero for a given polynomial"
Therefore,
x = -1
x + 1 = 0 , with multiplicity of two
x = 2
x - 2 = 0, with multiplicity one
x = 4
x - 4 = 0, with multiplicity one
The polynomial p(x) is given as:
p(x) is equal to product of zeros
[tex]p(x) = (x+1)^2(x-2)(x-4)[/tex]
[tex]p(x) = (x^2 + 2x + 1)(x-2)(x-4)\\\\Muliply\ each\ term\ in\ first\ bracket\ with\ each\ term\ in\ second\ bracket\\\\p(x) = (x^3 -2x^2 +2x^2 -4x + x - 2)(x-4)\\\\p(x) = (x^4-4x^3 -2x^3+8x^2 +2x^3 -8x^2 -4x^2 + 16x +x^2-4x -2x + 8)\\\\Simplify\ the\ above\\\\p(x) = x^4 -4x^3-3x^2+10x +8[/tex]
Thus the polynomial is found
Using the Factor Theorem, the polynomial is given by:
[tex]p(x) = x^4 - 4x^3 - 3x^2 + 10x + 8[/tex], which is option 3.
The Factor Theorem states that a polynomial function with roots is given by:
[tex]f(x) = a(x - x_1)(x - x_2)...(x - x_n)[/tex]
In which a is the leading coefficient.
In this problem:
- The zeroes are x = -1, with multiplicity 2, x = 2 and x = 4, hence [tex]x_1 = x_2 = -1, x_3 = 2, x_4 = 4[/tex].
- The leading coefficient is [tex]a = 1[/tex].
Hence, the polynomial is given by:
[tex]p(x) = (x + 1)^2(x - 2)(x - 4)[/tex]
[tex]p(x) = (x^2 + 2x + 1)(x^2 - 6x + 8)[/tex]
[tex]p(x) = x^4 - 4x^3 - 3x^2 + 10x + 8[/tex]
A similar problem is given at https://brainly.com/question/24380382
