Question

Use the factor theorem to determine whether v⁵ is a factor of v⁴ - 16v³ + 8v² - 725.

a) No
b) Yes

Answer 1

Using the Factor Theorem, we find that vµ is not a factor of the polynomial because substituting v = 0 into the polynomial results in -725, not 0.

Step-by-step explanation:

To determine whether vµ is a factor of v⁴ - 16v³ + 8v² - 725, we can use the Factor Theorem. The Factor Theorem states that if f(x) is a polynomial and f(a) = 0, then (x - a) is a factor of f(x). For vµ to be a factor of the polynomial, the remainder when the polynomial is divided by v must be zero.

This would imply that substituting v = 0 into the polynomial must result in a value of 0. Doing so, we get 0⁴ - 16·(0³) + 8·(0²) - 725 which simplifies to -725. Since -725 is not equal to 0, vµ is not a factor of the polynomial. Hence, the correct answer is: (a) No.

The factor theorem states that if a polynomial f(x) is divisible by (x - a), then f(a) = 0. In this case, we need to determine whether v^5 is a factor of v^4 - 16v^3 + 8v^2 - 725. To do this, we can substitute v = 0 into the polynomial and check if it equals 0:

f(0) = (0)^4 - 16(0)^3 + 8(0)^2 - 725 = -725

Since f(0) = -725 and it's not equal to 0, we can conclude that v^5 is not a factor of the polynomial. Therefore, the answer is a) No.