The polynomial function is P(x) = x⁴ - 14x³ + 59x² - 50x - 74
A polynomial is a function in which the least power of the unknown is 2.
The root of a polynomial is the value of the unknown which makes the polynomial equal zero.
Since we have that the given polynomial has roots 1 - √3 and 6 - i.
We also know that their conjugates 1 + √3 and 6 + i are also roots of the polynomial.
So, the factors of the polynomial are
So, the required polynomial is P(x) = [(x - 1) + √3][(x - 1) - √3][(x - 6) + i][(x - 6) - i]
= [(x - 1)² - (√3)²][(x - 6)² - i²] (Since a² - b² = (a - b)(a + b))
= [(x - 1)² - 3][(x - 6)² - (- 1)]
= [(x - 1)² - 3][(x - 6)² + 1]
Expanding the brackets, we have
= [x² - 2x + 1 - 3][(x² - 12x + 36 + 1]
= [x² - 2x - 2][(x² - 12x + 37]
= x⁴ - 12x³ + 37x² - 2x³ + 24x² - 74x - 2x² + 24x - 74
Collecting like terms, we have
= x⁴ - 14x³ + 59x² - 50x - 74
So, the polynomial is P(x) = x⁴ - 14x³ + 59x² - 50x - 74
Learn more about polynomial here:
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