Answer :
The roots of the polynomial f(x) = x⁴ + x³ + 23x² + 25x - 50 are: 1, -2, +5i and -5i
How to find the roots of the polynomial function?
The polynomial function is given as:
f(x) = x⁴ + x³ + 23x² + 25x - 50
Let us try x = 1:
f(1) = (1)⁴ + (1)³ + 23(1)² + 25(1) - 50
f(1) = 1 + 1 + 23 + 25 - 50
f(1) = 0
Thus, 1 is a root and the factor is (x - 1)
Now, let us try x = -2:
f(-2) = (-2)⁴ + (-2)³ + 23(-2)² + 25(-2) - 50
f(-2) = 16 - 8 + 92 - 50 - 50
f(-2) = 0
Thus, -2 is a root and the factor is (x + 2)
Multiplying both factors gives us:
(x - 1)(x + 2) = x² + x - 2
Thus, let us use long division to simplify the polynomial:
x² + 25
x² + x - 2 | x⁴ + x³ + 23x² + 25x - 50
- x⁴ + x³ - 2x²
25x² + 25x - 50
- 25x² + 25x - 50
0
Simplifying to find the remaining roots gives:
x² + 25 = 0
x² = -25
x = ±√-25
x = ±5i
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