To solve the problem of dividing the polynomial [tex]\(4x^2 + 3x - 1\)[/tex] by [tex]\(x - 2\)[/tex], we use polynomial long division. Here's the detailed step-by-step process:
1. Setup
[tex]\[
\text{Dividend: } 4x^2 + 3x - 1
\][/tex]
[tex]\[
\text{Divisor: } x - 2
\][/tex]
2. First Division Step:
- Divide the leading term of the dividend ([tex]\(4x^2\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]) to get the first term of the quotient:
[tex]\[
\frac{4x^2}{x} = 4x
\][/tex]
3. Multiply and Subtract:
- Multiply [tex]\(4x\)[/tex] by the divisor ([tex]\(x - 2\)[/tex]):
[tex]\[
4x \cdot (x - 2) = 4x^2 - 8x
\][/tex]
- Subtract this from the original polynomial:
[tex]\[
(4x^2 + 3x - 1) - (4x^2 - 8x) = 3x + 8x - 1 = 11x - 1
\][/tex]
4. Second Division Step:
- Now, divide the leading term of the new polynomial ([tex]\(11x\)[/tex]) by the leading term of the divisor ([tex]\(x\)[/tex]):
[tex]\[
\frac{11x}{x} = 11
\][/tex]
5. Multiply and Subtract:
- Multiply [tex]\(11\)[/tex] by the divisor ([tex]\(x - 2\)[/tex]):
[tex]\[
11 \cdot (x - 2) = 11x - 22
\][/tex]
- Subtract this from the current polynomial:
[tex]\[
(11x - 1) - (11x - 22) = -1 + 22 = 21
\][/tex]
Thus, the quotient is [tex]\(4x + 11\)[/tex] and the remainder is [tex]\(21\)[/tex].
So, the remainder when [tex]\(4x^2 + 3x - 1\)[/tex] is divided by [tex]\(x - 2\)[/tex] is [tex]\(\boxed{21}\)[/tex].
