To find the quotient and remainder of the expression [tex]\(\frac{2 x^3 + m x^2 + (3 m - 1) x + 5 m}{x + 1}\)[/tex] using polynomial long division, we follow these steps:
### Step 1: Set Up the Division
We want to divide [tex]\(2 x^3 + m x^2 + (3 m - 1) x + 5 m\)[/tex] by [tex]\(x + 1\)[/tex].
### Step 2: Divide the Leading Terms
- The leading term of the numerator is [tex]\(2x^3\)[/tex], and the leading term of the denominator is [tex]\(x\)[/tex].
- Divide the leading terms: [tex]\(\frac{2x^3}{x} = 2x^2\)[/tex].
### Step 3: Multiply and Subtract
- Multiply the entire divisor [tex]\(x + 1\)[/tex] by the result from Step 2:
[tex]\[
2x^2 \cdot (x + 1) = 2x^3 + 2x^2
\][/tex]
- Subtract this from the original polynomial:
[tex]\[
(2 x^3 + m x^2 + (3 m - 1) x + 5 m) - (2x^3 + 2x^2) = (m - 2)x^2 + (3m - 1)x + 5m
\][/tex]
### Step 4: Repeat the Process
- Divide the new leading term [tex]\((m - 2)x^2\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{(m - 2)x^2}{x} = (m - 2)x
\][/tex]
- Multiply the entire divisor [tex]\(x + 1\)[/tex] by [tex]\((m - 2)x\)[/tex]:
[tex]\[
(m - 2)x \cdot (x + 1) = (m - 2)x^2 + (m - 2)x
\][/tex]
- Subtract this from the current polynomial:
[tex]\[
((m - 2)x^2 + (3m - 1)x + 5m) - ((m - 2)x^2 + (m - 2)x) = (3m - m + 1)x + 5m = (2m + 1)x + 5m
\][/tex]
### Step 5: Repeat Again
- Divide the new leading term [tex]\((2m + 1)x\)[/tex] by [tex]\(x\)[/tex]:
[tex]\[
\frac{(2m + 1)x}{x} = 2m + 1
\][/tex]
- Multiply the entire divisor [tex]\(x + 1\)[/tex] by [tex]\(2m + 1\)[/tex]:
[tex]\[
(2m + 1) \cdot (x + 1) = (2m + 1)x + (2m + 1)
\][/tex]
- Subtract this from the current polynomial:
[tex]\[
((2m + 1)x + 5m) - ((2m + 1)x + (2m + 1)) = 5m - (2m + 1) = 3m - 1
\][/tex]
### Step 6: Identify the Quotient and Remainder
- The quotient is obtained by summing up the results from each division step: [tex]\(2x^2 + (m - 2)x + (2m + 1)\)[/tex].
- The remainder is the remaining term after the final subtraction: [tex]\(3m - 1\)[/tex].
Thus, the quotient is:
[tex]\[
\frac{2x^2}{x+1} + \frac{(m - 2)x}{x+1} + \frac{2m + 1}{x+1}
\][/tex]
And the remainder is:
[tex]\[
\frac{3m - 1}{x + 1}
\][/tex]
Putting it all together, the division results in:
[tex]\[
\left( \frac{2x^2}{x+1} + \frac{(m-2)x}{x+1} + \frac{2m+1}{x+1}, \frac{3m - 1}{x + 1} \right)
\][/tex]
