Certainly! Here is the detailed, step-by-step solution to find the remainder when dividing [tex]\( \frac{12x^3 - 8x^2 + 9x - 7}{2x - 1} \)[/tex] using polynomial long division:
### Step 1: Setup the Long Division
Arrange the polynomials for long division, where the numerator is the dividend [tex]\( 12x^3 - 8x^2 + 9x - 7 \)[/tex] and the denominator is the divisor [tex]\( 2x - 1 \)[/tex].
### Step 2: Perform the Division
First Division Step:
- Divide the leading term of the numerator [tex]\( 12x^3 \)[/tex] by the leading term of the denominator [tex]\( 2x \)[/tex].
- [tex]\[ \frac{12x^3}{2x} = 6x^2 \][/tex]
- Multiply [tex]\( 6x^2 \)[/tex] by [tex]\( 2x - 1 \)[/tex] and subtract from the original dividend:
[tex]\[ (12x^3 - 8x^2 + 9x - 7) - (6x^2 \cdot (2x - 1)) \][/tex]
[tex]\[ 12x^3 - 8x^2 + 9x - 7 - (12x^3 - 6x^2) \][/tex]
[tex]\[ (-8x^2 + 9x - 7) - (-6x^2) \][/tex]
[tex]\[ -8x^2 + 6x^2 + 9x - 7 = -2x^2 + 9x - 7 \][/tex]
Second Division Step:
- Divide the new leading term [tex]\( -2x^2 \)[/tex] by [tex]\( 2x \)[/tex].
- [tex]\[ \frac{-2x^2}{2x} = -x \][/tex]
- Multiply [tex]\( -x \)[/tex] by [tex]\( 2x - 1 \)[/tex] and subtract from the new dividend:
[tex]\[ (-2x^2 + 9x - 7) - (-x \cdot (2x - 1)) \][/tex]
[tex]\[ (-2x^2 + 9x - 7) - (-2x^2 + x) \][/tex]
[tex]\[ (-2x^2 + 9x - 7) - (-2x^2 + x) \][/tex]
[tex]\[ -2x^2 + 9x - 7 + 2x^2 - x = 8x - 7 \][/tex]
Third Division Step:
- Divide the new leading term [tex]\( 8x \)[/tex] by [tex]\( 2x \)[/tex].
- [tex]\[ \frac{8x}{2x} = 4 \][/tex]
- Multiply [tex]\( 4 \)[/tex] by [tex]\( 2x - 1 \)[/tex] and subtract from the new dividend:
[tex]\[ (8x - 7) - (4 \cdot (2x - 1)) \][/tex]
[tex]\[ (8x - 7) - (8x - 4) \][/tex]
[tex]\[ 8x - 7 - 8x + 4 = -3 \][/tex]
### Conclusion
The quotient obtained from the division is [tex]\( 6x^2 - x + 4 \)[/tex], and the remainder is [tex]\( -3 \)[/tex].
So, the remainder when [tex]\( \frac{12x^3 - 8x^2 + 9x - 7}{2x - 1} \)[/tex] is divided is [tex]\( -3 \)[/tex].
Answer: D. -3
