To find the remainder when the polynomial \(4x^2 + 10x - 4\) is divided by \(2x - 1\), we can use polynomial long division. Here are the detailed steps to solve the problem:
1. Set up the division:
- Dividend: \(4x^2 + 10x - 4\)
- Divisor: \(2x - 1\)
2. Divide the leading term of the dividend by the leading term of the divisor:
- Leading term of the dividend: \(4x^2\)
- Leading term of the divisor: \(2x\)
- \(\frac{4x^2}{2x} = 2x\)
3. Multiply the entire divisor by this result (the quotient term):
- \(2x \cdot (2x - 1) = 4x^2 - 2x\)
4. Subtract this from the original dividend:
[tex]\[
(4x^2 + 10x - 4) - (4x^2 - 2x) = 12x - 4
\][/tex]
5. Repeat the process with the new polynomial:
- Divide the leading term of the new polynomial by the leading term of the divisor:
\(\frac{12x}{2x} = 6\)
6. Multiply the entire divisor by this result:
- \(6 \cdot (2x - 1) = 12x - 6\)
7. Subtract this from the new polynomial:
[tex]\[
(12x - 4) - (12x - 6) = 2
\][/tex]
Thus, the remainder when [tex]\(4x^2 + 10x - 4\)[/tex] is divided by [tex]\(2x - 1\)[/tex] is [tex]\(\boxed{2}\)[/tex].
