Use the factor theorem to show that the first polynomial is not a factor of the second polynomial:

First Polynomial: [tex]x^5[/tex]
Second Polynomial: [tex]4x^4 - 21x^3 - 6x^2 - 4x - 15[/tex]

A) [tex]x^5[/tex] is a factor of [tex]4x^4 - 21x^3 - 6x^2 - 4x - 15[/tex] because the remainder is 0.
B) [tex]x[/tex] is not a factor of [tex]4x^4 - 21x^3 - 6x^2 - 4x - 15[/tex] because the remainder is not 0.

The polynomial x⁵ is not a factor of the polynomial 4x⁴ - 21x³ - 6x² - 4x - 15. We verified this firstly by substituting x = 0 into the second polynomial and not getting a remainder of 0 but -15. Secondly, the degree of the second polynomial is less than that of the first polynomial, providing further evidence that x⁵ is not a factor of the second polynomial.

Explanation:

To use the factor theorem to determine if x⁵ is a factor of 4x⁴ - 21x³ - 6x² - 4x - 15, we substitute x = 0 into the second polynomial.

So, 4(0)⁴ - 21(0)³ - 6(0)² - 4(0) - 15 = -15.

Since the remainder is not 0, according to the factor theorem, x⁵ is not a factor of 4x⁴ - 21x³ - 6x² - 4x - 15.

Alternatively, if the first polynomial x⁵ was a factor of the second polynomial, the degree of the second polynomial would be at least 5, which it is not in this case. Therefore, it also serves as another line of reasoning that x⁵ could not be a factor of 4x⁴ - 21x³ - 6x² - 4x - 15.

Learn more about Factor theorem here:

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