Find the polynomial function f with real coefficients that has given degree, zeros, and solution point.
Degree: 3
Zeros: -2,1- root of 2i
Solution point: f(-1)=-54
since complex solutions appear in conjugate pairs, the third solution must be 1 + √2 i
so the function has the form
f(x) = a(x+2)(x - 1 + √2 i)(x - 1 - √2 i)
= a(x-2)(x^2 - 2x + 3)
since f(-1) = a(-3)(1 + 2 + 3) = -18a
so -18a = -54
a = 3
f(x) = 3(x+2)(x^2 - 2x + 3)
To find the polynomial function f with the given degree, zeros, and solution point, we will use the fact that a polynomial function can be written as a product of its linear factors and a constant term.
Let's start with the zeros. We have three zeros: -2, 1-√2i, and 1+√2i. Since the coefficients are real, complex zeros occur in conjugate pairs. Therefore, we can combine the complex zeros into a quadratic factor.
The quadratic factor would be -[ (x - (1 - √2i))(x - (1 + √2i))] = -( (x - 1 + √2i)(x - 1 - √2i)).
Now let's write the linear factors for all the zeros. The linear factor for -2 would be (x - (-2)) = (x + 2).
Now we can write the polynomial function using these factors. By multiplying the quadratic and linear factors, we get:
f(x) = -( (x - 1 + √2i)(x - 1 - √2i))(x + 2).
To find the value of the constant term in the polynomial function, we can use the solution point f(-1) = -54. Substituting -1 for x in the function, we get:
-54 = -( (-1 - 1 + √2i)(-1 - 1 - √2i))( -1 + 2).
Simplifying this equation will allow us to find the constant term.
-54 = -( ( -2 + √2i)( -2 - √2i))(1).
Now we need to simplify the quadratic multiplication.
-54 = -( -2 * -2 + (-2 * √2i) + ( -2 * - √2i) + ( √2i * - √2i))(1).
Simplifying further:
-54 = -(4 - 2√2i + 2√2i - 2i^2)(1).
Now, we can simplify the equation:
-54 = -(4 - 2i^2)(1).
Since i^2 = -1, we can substitute:
-54 = -(4 - 2(-1))(1).
Simplifying the equation:
-54 = -(4 + 2)(1).
-54 = -(6)(1).
-54 = -6.
Therefore, we have a contradiction. The equation -54 = -6 is not true, so there is no polynomial function with the given degree, zeros, and solution point. Please double-check the given information or provide additional details if necessary.