Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Join our Q&A platform to connect with experts dedicated to providing precise answers to your questions in different areas. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
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]
Thanks for using our platform. We aim to provide accurate and up-to-date answers to all your queries. Come back soon. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
