A polynomial f (x) and one of its zeros are given. Factor f (x) as a product of linear factors.f (x) = 4x3 - 23x2 + 46x + 13; 3 + 2i is a zeroSelect one:a. (4x + 1)(x - (3 + 2i))(x + (3 + 2i))b. (4x - 1)(x - (3 + 2i))(x - (3 - 2i))c. (4x + 1)(x - (3 + 2i))(x - (3 - 2i))d. (4x - 1)(x - (3 + 2i))(x + (3 + 2i))
Question
A polynomial f (x) and one of its zeros are given. Factor f (x) as a product of linear factors.f (x) = 4x3 - 23x2 + 46x + 13; 3 + 2i is a zeroSelect one:a. (4x + 1)(x - (3 + 2i))(x + (3 + 2i))b. (4x - 1)(x - (3 + 2i))(x - (3 - 2i))c. (4x + 1)(x - (3 + 2i))(x - (3 - 2i))d. (4x - 1)(x - (3 + 2i))(x + (3 + 2i))
Solution
The correct answer is a. (4x + 1)(x - (3 + 2i))(x + (3 + 2i))
Here's why:
The polynomial f(x) = 4x^3 - 23x^2 + 46x + 13 has a given zero of 3 + 2i.
According to the Complex Conjugate Root Theorem, if a polynomial has a complex root a + bi, then its conjugate a - bi is also a root of the polynomial.
So, if 3 + 2i is a root, then 3 - 2i is also a root.
Therefore, the polynomial can be factored as (x - (3 + 2i))(x - (3 - 2i)) = (x - 3 - 2i)(x - 3 + 2i).
To find the third factor, we can divide the original polynomial by the product of these two factors.
Doing so gives us the third factor, 4x + 1.
So, the polynomial f(x) can be factored as (4x + 1)(x - (3 + 2i))(x + (3 + 2i)).
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