Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Discover in-depth solutions to your questions from a wide range of experts on our user-friendly Q&A platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
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]
### 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]
We appreciate your time on our site. Don't hesitate to return whenever you have more questions or need further clarification. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're dedicated to helping you find the answers you need at Westonci.ca. Don't hesitate to return for more.
