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Sagot :
Answer:
[tex]Q(x) = x^3 + 4x^2 - 10x + 12[/tex]
Step-by-step explanation:
Complex numbers:
The most important relation when dealing with complex numbers is that:
[tex]i^2 = -1[/tex]
Zeros of a function:
Given a polynomial f(x), this polynomial has roots [tex]x_{1}, x_{2}, x_{n}[/tex] such that it can be written as: [tex]a(x - x_{1})*(x - x_{2})*...*(x-x_n)[/tex], in which a is the leading coefficient.
Q has degree 3 and zeros −6 and 1 + i.
Complex roots are always complex-conjugate, which means that if 1 + i is a root, 1 - i is also a root.
I am going to say that the leading coefficient is 1. So
[tex]Q(x) = (x - (-6))(x - (1 + i))(x - (1 - i)) = (x + 6)(x^2 - x(1-i) - x(1+i) + (1+i)(1-i)) = (x + 6)(x^2 - 2x + 1 - i^2) = (x + 6)(x^2 - 2x + 1 - (-1)) = (x + 6)(x^2 - 2x + 2) = x^3 + 6x^2 - 2x^2 - 12x + 2x + 12 = x^3 + 4x^2 - 10x + 12[/tex]
The polynomial is:
[tex]Q(x) = x^3 + 4x^2 - 10x + 12[/tex]
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