Answer :
Certainly! Let's divide the polynomial \(\ell x^2 + mx + n\) by \(x + 1\).
### Step-by-Step Solution:
1. Setup the Division:
We need to divide the polynomial \(\ell x^2 + mx + n\) by \(x + 1\).
2. Perform Polynomial Division (using long division or synthetic division):
We start by dividing the first term of the polynomial (\(\ell x^2\)) by the first term of the divisor (\(x\)).
[tex]\[ \text{Quotient term: } \frac{\ell x^2}{x} = \ell x \][/tex]
3. Multiply and Subtract:
Next, we multiply the entire divisor \(x + 1\) by the quotient term \(\ell x\).
[tex]\[ (\ell x) \cdot (x + 1) = \ell x^2 + \ell x \][/tex]
Now, subtract \(\ell x^2 + \ell x\) from the original polynomial \(\ell x^2 + mx + n\):
[tex]\[ (\ell x^2 + mx + n) - (\ell x^2 + \ell x) = mx + n - \ell x = (\ell x - \ell x + m - m + n - n) + (m - \ell)x + n = (m - \ell)x + n \][/tex]
4. Repeat the Process:
Now we are left with the polynomial \((m - \ell)x + n\). We repeat the division process using this new polynomial (the remainder from the previous step).
Divide the first term \((m - \ell)x\) by \(x\):
[tex]\[ \text{New Quotient term: } \frac{(m - \ell)x}{x} = m - \ell \][/tex]
5. Combine Quotients:
Combining the quotient terms from both steps, we get:
[tex]\[ \text{Complete Quotient: } \ell x \][/tex]
6. Determine the Remainder:
Multiply the latest divisor \((x + 1)\) by the new quotient term \((m - \ell)\):
[tex]\[ ((m - \ell) \cdot (x + 1)) = (m - \ell)x + (m - \ell) \][/tex]
Subtract this from our current dividend:
[tex]\[ ((m - \ell)x + n) - ((m - \ell)x + (m - \ell)) = n - (m - \ell) = l - m + n \][/tex]
### Conclusion:
After performing the division:
- The Quotient is: \(\ell x - \ell + m\)
- The Remainder is: \(l - m + n\)
Thus, when [tex]\(\ell x^2 + mx + n\)[/tex] is divided by [tex]\(x + 1\)[/tex], the quotient is [tex]\(\ell x - \ell + m\)[/tex] and the remainder is [tex]\(l - m + n\)[/tex].
### Step-by-Step Solution:
1. Setup the Division:
We need to divide the polynomial \(\ell x^2 + mx + n\) by \(x + 1\).
2. Perform Polynomial Division (using long division or synthetic division):
We start by dividing the first term of the polynomial (\(\ell x^2\)) by the first term of the divisor (\(x\)).
[tex]\[ \text{Quotient term: } \frac{\ell x^2}{x} = \ell x \][/tex]
3. Multiply and Subtract:
Next, we multiply the entire divisor \(x + 1\) by the quotient term \(\ell x\).
[tex]\[ (\ell x) \cdot (x + 1) = \ell x^2 + \ell x \][/tex]
Now, subtract \(\ell x^2 + \ell x\) from the original polynomial \(\ell x^2 + mx + n\):
[tex]\[ (\ell x^2 + mx + n) - (\ell x^2 + \ell x) = mx + n - \ell x = (\ell x - \ell x + m - m + n - n) + (m - \ell)x + n = (m - \ell)x + n \][/tex]
4. Repeat the Process:
Now we are left with the polynomial \((m - \ell)x + n\). We repeat the division process using this new polynomial (the remainder from the previous step).
Divide the first term \((m - \ell)x\) by \(x\):
[tex]\[ \text{New Quotient term: } \frac{(m - \ell)x}{x} = m - \ell \][/tex]
5. Combine Quotients:
Combining the quotient terms from both steps, we get:
[tex]\[ \text{Complete Quotient: } \ell x \][/tex]
6. Determine the Remainder:
Multiply the latest divisor \((x + 1)\) by the new quotient term \((m - \ell)\):
[tex]\[ ((m - \ell) \cdot (x + 1)) = (m - \ell)x + (m - \ell) \][/tex]
Subtract this from our current dividend:
[tex]\[ ((m - \ell)x + n) - ((m - \ell)x + (m - \ell)) = n - (m - \ell) = l - m + n \][/tex]
### Conclusion:
After performing the division:
- The Quotient is: \(\ell x - \ell + m\)
- The Remainder is: \(l - m + n\)
Thus, when [tex]\(\ell x^2 + mx + n\)[/tex] is divided by [tex]\(x + 1\)[/tex], the quotient is [tex]\(\ell x - \ell + m\)[/tex] and the remainder is [tex]\(l - m + n\)[/tex].
