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2. Andrew made an error in determining the polynomial equation of smallest degree whose roots are 3, 2+2i and .

Review Andrew’s work, identify the error and correct all work from that point forward that is affected by this error. Include at least one sentence explaining his error. Use proper mathematical vocabulary, appropriately
(x-3)(x+2+2i)(x+2-2i)=0
(x-3)(x^2+4x+8)=0
x^3+x^2-4x-24=0)

Respuesta :

Using the Factor Theorem, Andrew's error was at the application of the minus signal to the complex roots, and the polynomial of degree 3 is of f(x) = x³ - 7x² + 20x - 24.

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

In which a is the leading coefficient.

For this problem, the roots are given as follows:

  • [tex]x_1 = 3[/tex].
  • [tex]x_2 = 2 + 2i[/tex].
  • [tex]x_3 = 2 - 2i[/tex].

Hence the polynomial is given as follows, considering leading coefficient a = 1:

[tex]f(x) = (x - 3)(x - 2 - 2i)(x - 2 + 2i)[/tex]

His error was at the application of the minus signal to the complex roots.

Then:

f(x) = (x - 3)[(x - 2)² - (2i)²]

f(x) = (x - 3)(x² - 4x + 4 + 4)

f(x) = (x - 3)(x² - 4x + 8)

f(x) = x³ - 7x² + 20x - 24.

More can be learned about the Factor Theorem at https://brainly.com/question/24380382

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