To solve this problem, we divide the polynomial [tex]\( p(x) = 4x^3 - 12x^2 + 11x - 5 \)[/tex] by [tex]\( 2x - 1 \)[/tex].
### Step-by-Step Division:
1. Setup the division: [tex]\(p(x)\)[/tex] is divided by [tex]\(2x - 1\)[/tex]. We perform polynomial long division.
2. Find the first term of the quotient: Divide the leading term of [tex]\( p(x) \)[/tex], which is [tex]\( 4x^3 \)[/tex], by the leading term of the divisor, [tex]\( 2x \)[/tex]:
[tex]\[
\frac{4x^3}{2x} = 2x^2
\][/tex]
This is the first term of the quotient.
3. Multiply: Multiply [tex]\( 2x^2 \)[/tex] (the first term of the quotient) by [tex]\( 2x - 1 \)[/tex]:
[tex]\[
2x^2 \cdot (2x - 1) = 4x^3 - 2x^2
\][/tex]
4. Subtract: Subtract this result from [tex]\( p(x) \)[/tex]:
[tex]\[
(4x^3 - 12x^2 + 11x - 5) - (4x^3 - 2x^2) = -10x^2 + 11x - 5
\][/tex]
5. Second term of the quotient: Divide the new leading term [tex]\( -10x^2 \)[/tex] by [tex]\( 2x \)[/tex]:
[tex]\[
\frac{-10x^2}{2x} = -5x
\][/tex]
This is the second term of the quotient.
6. Multiply: Multiply [tex]\( -5x \)[/tex] by [tex]\( 2x - 1 \)[/tex]:
[tex]\[
-5x \cdot (2x - 1) = -10x^2 + 5x
\][/tex]
7. Subtract: Subtract this result from the remainder:
[tex]\[
(-10x^2 + 11x - 5) - (-10x^2 + 5x) = 6x - 5
\][/tex]
8. Third term of the quotient: Divide the new leading term [tex]\( 6x \)[/tex] by [tex]\( 2x \)[/tex]:
[tex]\[
\frac{6x}{2x} = 3
\][/tex]
This is the third term of the quotient.
9. Multiply: Multiply [tex]\( 3 \)[/tex] by [tex]\( 2x - 1 \)[/tex]:
[tex]\[
3 \cdot (2x - 1) = 6x - 3
\][/tex]
10. Subtract: Subtract this result from the remainder:
[tex]\[
(6x - 5) - (6x - 3) = -2
\][/tex]
The quotient of the division is [tex]\(2x^2 - 5x + 3\)[/tex], and the remainder is [tex]\(-2\)[/tex].
Thus, the remainder when [tex]\( p(x) = 4x^3 - 12x^2 + 11x - 5 \)[/tex] is divided by [tex]\( 2x - 1 \)[/tex] is [tex]\(-2\)[/tex].
Therefore, the correct answer is:
[tex]\[
\boxed{-2}
\][/tex]
