Answer :
Sure, let's divide the polynomial \(-v + 11\) by \(v - 9\).
### Step-by-Step Solution:
1. Identify the terms in both the divisor and the dividend:
- Dividend (numerator): \(-v + 11\)
- Divisor (denominator): \(v - 9\)
2. Set up the division:
[tex]$[tex]$\frac{-v + 11}{v - 9}$[/tex]$[/tex]
3. Divide the first term of the numerator by the first term of the denominator:
- First term of the numerator: \(-v\)
- First term of the denominator: \(v\)
- Divide: \(\frac{-v}{v} = -1\)
4. Multiply the entire divisor by the result from Step 3:
- Result from Step 3: \(-1\)
- Multiply by the divisor:
[tex]\[ -1 \cdot (v - 9) = -v + 9 \][/tex]
5. Subtract this result from the original numerator:
- Original numerator: \(-v + 11\)
- Result from multiplication: \(-v + 9\)
[tex]\[ (-v + 11) - (-v + 9) = -v + 11 + v - 9 = 2 \][/tex]
6. Determine the quotient and the remainder from this process:
- The quotient (result from Step 3): \(-1\)
- The remainder (result from Step 5): \(2\)
So, the quotient when dividing \(-v + 11\) by \(v - 9\) is \(-1\) and the remainder is \(2\).
### Final Answer:
[tex]\[ \text{Quotient: } -1 \][/tex]
[tex]\[ \text{Remainder: } 2 \][/tex]
The division of [tex]\(-v + 11\)[/tex] by [tex]\(v - 9\)[/tex] gives a quotient of [tex]\(-1\)[/tex] and a remainder of [tex]\(2\)[/tex].
### Step-by-Step Solution:
1. Identify the terms in both the divisor and the dividend:
- Dividend (numerator): \(-v + 11\)
- Divisor (denominator): \(v - 9\)
2. Set up the division:
[tex]$[tex]$\frac{-v + 11}{v - 9}$[/tex]$[/tex]
3. Divide the first term of the numerator by the first term of the denominator:
- First term of the numerator: \(-v\)
- First term of the denominator: \(v\)
- Divide: \(\frac{-v}{v} = -1\)
4. Multiply the entire divisor by the result from Step 3:
- Result from Step 3: \(-1\)
- Multiply by the divisor:
[tex]\[ -1 \cdot (v - 9) = -v + 9 \][/tex]
5. Subtract this result from the original numerator:
- Original numerator: \(-v + 11\)
- Result from multiplication: \(-v + 9\)
[tex]\[ (-v + 11) - (-v + 9) = -v + 11 + v - 9 = 2 \][/tex]
6. Determine the quotient and the remainder from this process:
- The quotient (result from Step 3): \(-1\)
- The remainder (result from Step 5): \(2\)
So, the quotient when dividing \(-v + 11\) by \(v - 9\) is \(-1\) and the remainder is \(2\).
### Final Answer:
[tex]\[ \text{Quotient: } -1 \][/tex]
[tex]\[ \text{Remainder: } 2 \][/tex]
The division of [tex]\(-v + 11\)[/tex] by [tex]\(v - 9\)[/tex] gives a quotient of [tex]\(-1\)[/tex] and a remainder of [tex]\(2\)[/tex].
