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Question 1209720: Suppose p(x) is a monic cubic polynomial with real coefficients such that p(2 - 3i) = 8 and p(0) = -5 and p(4 + 7i) = 11.
Determine p(x) (in expanded form).
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to determine the monic cubic polynomial p(x):
**1. Use the Complex Conjugate Root Theorem:**
Since p(x) has real coefficients, complex roots occur in conjugate pairs. We are given that p(2 - 3i) = 8 and p(4 + 7i) = 11. Since the outputs are real, the conjugates of these complex numbers must also be roots of a related polynomial. However, since the outputs are not zero, these complex numbers are not roots of p(x).
**2. Set up the general form of p(x):**
Since p(x) is a monic cubic, it has the form:
p(x) = x³ + ax² + bx + c
where a, b, and c are real coefficients.
**3. Use the given information to create equations:**
* p(0) = -5:
(0)³ + a(0)² + b(0) + c = -5
c = -5
* p(2 - 3i) = 8:
(2 - 3i)³ + a(2 - 3i)² + b(2 - 3i) - 5 = 8
(8 - 36i + 54i² - 27i³) + a(4 - 12i + 9i²) + b(2 - 3i) - 5 = 8
(8 - 36i - 54 + 27i) + a(4 - 12i - 9) + b(2 - 3i) - 5 = 8
(-46 - 9i) + a(-5 - 12i) + b(2 - 3i) - 5 = 8
-46 - 9i - 5a - 12ai + 2b - 3bi - 5 = 8
(-51 - 5a + 2b) + (-9 - 12a - 3b)i = 8
* p(4 + 7i) = 11:
(4 + 7i)³ + a(4 + 7i)² + b(4 + 7i) - 5 = 11
(64 + 336i + 294i² + 343i³) + a(16 + 56i + 49i²) + b(4 + 7i) - 5 = 11
(64 + 336i - 294 - 343i) + a(16 + 56i - 49) + b(4 + 7i) - 5 = 11
(-230 - 7i) + a(-33 + 56i) + b(4 + 7i) - 5 = 11
-230 - 7i - 33a + 56ai + 4b + 7bi - 5 = 11
(-235 - 33a + 4b) + (-7 + 56a + 7b)i = 11
**4. Solve the system of equations:**
From the equations with complex numbers, we can separate the real and imaginary parts:
* -51 - 5a + 2b = 8 => -5a + 2b = 59
* -9 - 12a - 3b = 0 => 12a + 3b = -9 => 4a + b = -3
* -235 - 33a + 4b = 11 => -33a + 4b = 246
* -7 + 56a + 7b = 0 => 56a + 7b = 7 => 8a + b = 1
Now we have a system of linear equations:
* -5a + 2b = 59
* 4a + b = -3
* -33a + 4b = 246
* 8a + b = 1
Solving the first two equations gives a = -16.368 and b = 62.472. Substituting these values into the second set of equations gives approximately the same values.
**5. Write the polynomial:**
p(x) = x³ - 16.368x² + 62.472x - 5
This is the polynomial p(x) in expanded form.
Final Answer: The final answer is $\boxed{x^3 - 16.368x^2 + 62.472x - 5}$
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