Find a polynomial function with -2 and 3-i.
I will assume that these are the zeros or the solutions to
your function = 0
if 3-i is a zero, then 3+i is a zero, since complex solutions always appear in conjugate pairs
sum of those roots = 6
product of those roots = 9 - i^2 = 10
so the quadratic producing those two complex roots
is x^2 - 6x + 10
since -2 is a zero, then (x+2) must be a factor
a possible polynomial function is
f(x) = (x+2)(x^2 - 6x + 10)
expand if you feel like it
To find a polynomial function with -2 and 3-i as roots, we need to write the function in factored form using the root theorem.
Since 3-i is a root, we know that its conjugate, 3+i, must also be a root. Therefore, the factors of the polynomial are (x - (-2)), (x - (3 - i)), and (x - (3 + i)).
Simplifying these factors, we get (x + 2), (x - 3 + i), and (x - 3 - i).
Now, we can multiply these factors together to find the polynomial function:
(x + 2)(x - 3 + i)(x - 3 - i)
Multiplying the first two factors using the distributive property, we get:
(x + 2)(x - 3) + (x + 2)(i)
Expanding the first part and distributing the i, we have:
(x^2 - x - 6) + (xi + 2i)
Simplifying further:
x^2 - x - 6 + xi + 2i
Finally, combining like terms, we have:
x^2 - x - 6 + xi + 2i
Therefore, the polynomial function with -2 and 3-i as roots is:
f(x) = x^2 - x - 6 + xi + 2i (where i is the imaginary unit)
To find a polynomial function with the given roots, we need to use the concept of conjugate pairs. Let's understand how to do that step by step:
Step 1: Write the factors using the given roots:
- Root 1: -2
- Root 2: 3-i
Step 2: Find the conjugate of the second root:
To find the conjugate of 3-i, we simply change the sign of the imaginary part, i.e., 3-i becomes 3+i.
Step 3: Write the factors using the conjugate of the second root:
- Root 1: -2
- Root 2: 3-i
- Root 3: 3+i
Step 4: Multiply all the factors:
(x + 2)(x - (3 - i))(x - (3 + i))
Step 5: Simplify the multiplication:
(x + 2)[(x - 3 + i)(x - 3 - i)]
(x + 2)[(x^2 - 3x - ix - 3x + 9 + i3 - ix + i(3) + i^2)]
(x + 2)[(x^2 - 6x + 9 - 2i)]
Step 6: Expand and simplify further:
(x + 2)(x^2 - 6x + 9 - 2i)
x(x^2 - 6x + 9 - 2i) + 2(x^2 - 6x + 9 - 2i)
x^3 - 6x^2 + 9x - 2ix + 2x^2 - 12x + 18 - 4i
Step 7: Combine like terms to get the final polynomial function:
x^3 - 4x^2 - 3x - 12x + 18 - 2ix - 4i
x^3 - 4x^2 - 15x + 18 - 2ix - 4i
Therefore, a polynomial function with roots -2 and 3-i is:
f(x) = x^3 - 4x^2 - 15x + 18 - 2ix - 4i