To find the remainder of the polynomial 6x² + 11x + 7 when divided by 2x + 1, we can use the Remainder Theorem. According to this theorem, the remainder of the division of a polynomial f(x) by a linear divisor ax + b can be found by evaluating f(-b/a).
In our case, the divisor is 2x + 1, which means we need to find -b/a. Here, a = 2 and b = 1. Thus, we calculate:
-1/2
Next, we will substitute x = -1/2 into the polynomial 6x² + 11x + 7:
f(-1/2) = 6(-1/2)² + 11(-1/2) + 7
Calculating each term, we get:
- 6(-1/2)² = 6(1/4) = 3
- 11(-1/2) = -11/2
- 7 = 7
Now, summing these values:
f(-1/2) = 3 – 11/2 + 7
To combine these, we convert 7 into halves:
7 = 14/2
So:
f(-1/2) = 3 – 11/2 + 14/2 = 3 + 3/2 = 3 + 1.5 = 4.5
Thus, the remainder when the polynomial 6x² + 11x + 7 is divided by 2x + 1 is 4.5.