One factor of the polynomial 12X² - 17X - 5 is 4X + 1. What is the other factor?

To find the other factor of the polynomial (12X^2 - 17X - 5) when one factor is (4X + 1), we can use polynomial division or factorization methods.

Begin by recognizing that if (4X + 1) is a factor, the polynomial can be expressed as:

[

12X^2 - 17X - 5 = (4X + 1)(\text{other factor}).

]

To find the other factor, we perform polynomial long division or synthetic division of (12X^2 - 17X - 5) by (4X + 1).

1. Divide the leading term (12X^2) by the leading term (4X) to get (3X).

2. Multiply (3X) by (4X + 1) resulting in (12X^2 + 3X).

3. Subtract this from (12X^2 - 17X - 5):

[

(12X^2 - 17X - 5) - (12X^2 + 3X) = -20X - 5.

\