Posted by Anon on Friday, May 24, 2013 at 5:49am.
Create a 3rd degree polynomial with real coefficients that has roots -1 and 4i. Write your answer in form ax^3 + bx^2 + cx+ d .
- Math - Bosnian, Friday, May 24, 2013 at 10:01am
The general equation for the function is:
f( x ) = a ( x - x1 ) ( x - x2 )( x - x3 )
where x1, x2 and x3 are the roots and a is the coefficient of x ^ 3
There is a theorem in algebra called the Imaginary Zeros Theorem.
It says that for a polynomial that is restricted to real coefficients then for any imaginary zero (root) that exists then its conjugate also exists.
This means that the roots you gave :
-1 and ( 4 + i ), also includes the congugate root:
( 4 - i )
This is the conjugate to ( 4 + i )
Since there are no other restrictions to the polynomial you can construct the roots in factored form like this:
f ( x ) = a [ x - ( - 1 ) ] ( x - 4 i ) ( x + 4 i ) =
a ( x + 1 ) ( x - 4 i ) ( x + 4 i ) =
a ( x + 1 ) ( x * x - x * 4 i + 4 i * x + 4 i * ( - 4 i ) =
a ( x + 1 ) ( x ^ 2 - 4 i x + 4 i x - 16 i ^ 2 ) =
a ( x + 1 ) [ x ^ 2 - 4 i x + 4 i x - 16 * ( - 1 ) ] =
a ( x + 1 ) ( x ^ 2 + 16 ) =
a ( x ^ 2 * x + x ^ 2 * 1 + 16 * x + 16 * 1 ) =
a ( x ^ 3 + x ^ 2 + 16 x + 16 )
a can be any real number.
If a = 1 then
f ( x ) = x ^ 3 + x ^ 2 + 16 x + 16
Answer this Question
More Related Questions
- Math - Create a 3rd degree polynomial with real coefficients that has roots -1 ...
- Math - A polynomial function f(x) has degree 6 and has real coefficients. It is ...
- calculus--please help!! - I have two questions that I don't understand and need ...
- MATH - write, in extended form, a polynomial(with real coefficients) of degree 3...
- Math (Algebra) - For each degree 17 polynomial f with real coefficients, let sf ...
- pre cal - Write a polynomial function of minimum degree with real coefficients ...
- Algebra - Write a polynomial of least degree with real coefficients and with the...
- Math - Fundamental Theorem - We can actually use the Zeros Theorem and the ...
- Form a polynomial f(x) from coefficient and it's 0 - Form a polynomial, f(x), ...
- Algebra - Can someone please explain how to do these problems. 1)write a ...