To find the quotient and the remainder when dividing the polynomial [tex]\(6x^2 - 7x + s\)[/tex] by [tex]\(2x + 1\)[/tex], follow these steps:
### Step-by-Step Solution:
1. Setup the Division:
- We want to divide [tex]\(6x^2 - 7x + s\)[/tex] by [tex]\(2x + 1\)[/tex].
2. First Division Step:
- Divide the leading term of the dividend [tex]\(6x^2\)[/tex] by the leading term of the divisor [tex]\(2x\)[/tex]:
[tex]\[
\frac{6x^2}{2x} = 3x
\][/tex]
- Multiply [tex]\(3x\)[/tex] by the divisor [tex]\(2x + 1\)[/tex]:
[tex]\[
3x \cdot (2x + 1) = 6x^2 + 3x
\][/tex]
- Subtract this result from the original polynomial:
[tex]\[
(6x^2 - 7x + s) - (6x^2 + 3x) = -7x - 3x + s = -10x + s
\][/tex]
3. Second Division Step:
- Now divide the new leading term [tex]\(-10x\)[/tex] by the leading term of the divisor [tex]\(2x\)[/tex]:
[tex]\[
\frac{-10x}{2x} = -5
\][/tex]
- Multiply [tex]\(-5\)[/tex] by the divisor [tex]\(2x + 1\)[/tex]:
[tex]\[
-5 \cdot (2x + 1) = -10x - 5
\][/tex]
- Subtract this result from the current polynomial:
[tex]\[
(-10x + s) - (-10x - 5) = s - (-5) = s + 5
\][/tex]
4. Conclusion:
- The quotient (result from the division) is [tex]\(3x - 5\)[/tex].
- The remainder is [tex]\(s + 5\)[/tex].
Thus, the quotient is [tex]\(3x - 5\)[/tex] and the remainder is [tex]\(s + 5\)[/tex].
