Respuesta :
Answer: x^4 + 2x³ + 10x² + 18x + 9
Step-by-step explanation:
From the Factor Theorem, a 4th degree polynomial has exactly 4 solutions (or 4 'zeros') across the set of complex numbers.
We are told that (-1) is a zero twice so that takes care of 2 out of 4.
We are also told that 3i is a zero.
For polynomials with real-number coefficients, non-real solutions always come in conjugate pairs. Thus, if 3i is a zero, then (-3i) is also a zero. We now have a total of 4 zeros: -1, -1, 3i, and -3i.
Also from the Factor Theorem, we can write a polynomial P(x) as the product of a series of binomial factors of the form (x-c), where each 'c' is a zero of the polynomial.
Now that we know the 3 items underlined above, and we have our 4 zeros, we can get started creating the 4th-degree polynomial.
P(x) = [x - (-1)] * [x - (-1)] * [x - 3i] * [x - (-3i)] = 0
P(x) = (x + 1) (x + 1) (x - 3i) (x + 3i) = 0
P(x) = (x² + 2x + 1) (x² - 9i²)
Since i² = -1, we can write
P(x) = (x² + 2x + 1) [x² - 9(-1)]
P(x) = (x² + 2x + 1) (x² + 9)
P(x) = x²(x² + 9) + 2x(x² + 9) + 1(x² + 9)
P(x) = x^4 + 9x² + 2x³ + 18x + x² + 9
P(x) = x^4 + 2x³ + (9x² + x²) + 18x + 9
Answer --> P(x) = x^4 + 2x³ + 10x² + 18x + 9
I hope this helps!
