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10) f(x) = x5 - 10x4 + 42x3 -124 x2 + 297x - 306; zero: 3i ? A) 2, -3i, -4 - i, -4 + i C) 2, -3i, 4 - i, 4 + i B) -2, -3i, -4 -i, -4 + i D) -2, -3i, 4-i, 4 + i

User Sowiarz
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1 Answer

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15 votes

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!!!

User Zahlii
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