Answer
Option C is correct.
The roots of the given function include
2, -3i, (4 + i), (4 - i)
Step-by-step explanation
To solve this, we would put the given roots of the solution into the place of x. The ones that give 0 are the roots of the expression
The expression is
f(x) = x⁵ - 10x⁴ + 42x³ - 124x² + 297x - 306
Starting with 2
f(x) = x⁵ - 10x⁴ + 42x³ - 124x² + 297x - 306
f(2) = 2⁵ - 10(2)⁴ + 42(2)³ - 124(2)² + 297(2) - 306
= 32 - 160 + 336 - 496 + 594 - 306
= 0
So, 2 is a root
-3i
f(x) = x⁵ - 10x⁴ + 42x³ - 124x² + 297x - 306
f(-3i) = (-3i)⁵ - 10(-3i)⁴ + 42(-3i)³ - 124(-3i)² + 297(-3i) - 306
= -243i - 810 + 1134i - 1116 - 891i - 306
= 0
So, -3i is also a root
4 + i
f(x) = x⁵ - 10x⁴ + 42x³ - 124x² + 297x - 306
f(4 + i) = (4 + i)⁵ - 10(4 + i)⁴ + 42(4 + i)³ - 124(4 + i)² + 297(4 + i) - 306
= 0
So, we know that the right root, when inserted and expanded will reduce the expression to 0.
4 - i
f(x) = x⁵ - 10x⁴ + 42x³ - 124x² + 297x - 306
f(4 - i) = (4 - i)⁵ - 10(4 - i)⁴ + 42(4 - i)³ - 124(4 - i)² + 297(4 - i) - 306
= 0
Inserting any of the other answers will result in answers other than 0 and show that they aren't roots/zeros for this expression.
Hope this Helps!!!