Final answer:
To find the value of p, we set x=-1/2 as the root of the factor (2x + 1) and substitute it into the polynomial 6x^2 + px + 4. After simplifying, we find that p equals 11.
Step-by-step explanation:
To find the value of p when (2x + 1) is a factor of the polynomial 6x^2 + px + 4, we can use polynomial division or apply the factor theorem. The factor theorem states that if (2x + 1) is a factor of 6x^2 + px + 4, then 6x^2 + px + 4 is zero when x equates to the root provided by the factor. In this case, the root is -1/2 (setting 2x + 1 = 0 and solving for x). Substituting -1/2 into the polynomial gives us:
6(-1/2)^2 + p(-1/2) + 4 = 0
Expanding and simplifying the equation:
6(1/4) - p/2 + 4 = 0
3/2 - p/2 + 4 = 0
Adding p/2 to both sides and subtracting 3/2 + 4 (which is 11/2) from both sides, we get:
p/2 = 11/2
Multiplying both sides by 2 yields:
p = 11
Therefore, the value of p is 11.