Find answers to your questions faster and easier with IDNLearn.com. Discover prompt and accurate answers from our experts, ensuring you get the information you need quickly.
Sagot :
To solve the polynomial division [tex]\( (2x^3 + 4x^2 - 35x + 15) \div (x-3) \)[/tex] using synthetic division, we proceed as follows:
1. Identify the coefficients of the polynomial:
The polynomial [tex]\( 2x^3 + 4x^2 - 35x + 15 \)[/tex] has coefficients [tex]\([2, 4, -35, 15]\)[/tex].
2. Determine the root for synthetic division:
For the divisor [tex]\( (x-3) \)[/tex], the root is [tex]\( x = 3 \)[/tex].
3. Set up the synthetic division table:
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & & 6 & 30 & -15 \\ \hline & 2 & 10 & -5 & 0 \\ \end{array} \][/tex]
4. Start with the leading coefficient:
The first coefficient is [tex]\(2\)[/tex], which goes directly into the row below the line.
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & 2 & & & \\ \hline & 2 & & & \\ \end{array} \][/tex]
5. Multiply and add:
- Multiply the leading coefficient [tex]\(2\)[/tex] by the root [tex]\(3\)[/tex] and place the result under the next coefficient [tex]\(4\)[/tex]:
[tex]\[2 \times 3 = 6\][/tex]
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & 2 & 6 & & \\ \hline & 2 & 10& & \\ \end{array} \][/tex]
- Add this result ([tex]\(6\)[/tex]) to the next coefficient ([tex]\(4\)[/tex]):
[tex]\[4 + 6 = 10\][/tex]
6. Repeat the process:
- Multiply the new result ([tex]\(10\)[/tex]) by [tex]\(3\)[/tex] and place it under the next coefficient ([tex]\(-35\)[/tex]):
[tex]\[10 \times 3 = 30\][/tex]
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & 2 & 6 & 30 & \\ \hline & 2 & 10& -5 & \\ \end{array} \][/tex]
- Add the new result ([tex]\(30\)[/tex]) to the next coefficient ([tex]\(-35\)[/tex]):
[tex]\[-35 + 30 = -5\][/tex]
- Multiply the new result ([tex]\(-5\)[/tex]) by [tex]\(3\)[/tex] and place it under the next coefficient ([tex]\(15\)[/tex]):
[tex]\[-5 \times 3 = -15\][/tex]
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & 2 & 6 & 30 & -15 \\ \hline & 2 & 10& -5 & 0 \\ \end{array} \][/tex]
- Add this result ([tex]\(-15\)[/tex]) to the last coefficient ([tex]\(15\)[/tex]):
[tex]\[15 + (-15) = 0\][/tex]
7. Result of synthetic division:
The quotient is given by the numbers we have in the last row except the last term, and the remainder is the last term in the last row.
- Quotient: [tex]\[ 2x^2 + 10x - 5 \][/tex]
- Remainder: [tex]\[ 0 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{2x^2 + 10x - 5} \][/tex]
1. Identify the coefficients of the polynomial:
The polynomial [tex]\( 2x^3 + 4x^2 - 35x + 15 \)[/tex] has coefficients [tex]\([2, 4, -35, 15]\)[/tex].
2. Determine the root for synthetic division:
For the divisor [tex]\( (x-3) \)[/tex], the root is [tex]\( x = 3 \)[/tex].
3. Set up the synthetic division table:
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & & 6 & 30 & -15 \\ \hline & 2 & 10 & -5 & 0 \\ \end{array} \][/tex]
4. Start with the leading coefficient:
The first coefficient is [tex]\(2\)[/tex], which goes directly into the row below the line.
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & 2 & & & \\ \hline & 2 & & & \\ \end{array} \][/tex]
5. Multiply and add:
- Multiply the leading coefficient [tex]\(2\)[/tex] by the root [tex]\(3\)[/tex] and place the result under the next coefficient [tex]\(4\)[/tex]:
[tex]\[2 \times 3 = 6\][/tex]
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & 2 & 6 & & \\ \hline & 2 & 10& & \\ \end{array} \][/tex]
- Add this result ([tex]\(6\)[/tex]) to the next coefficient ([tex]\(4\)[/tex]):
[tex]\[4 + 6 = 10\][/tex]
6. Repeat the process:
- Multiply the new result ([tex]\(10\)[/tex]) by [tex]\(3\)[/tex] and place it under the next coefficient ([tex]\(-35\)[/tex]):
[tex]\[10 \times 3 = 30\][/tex]
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & 2 & 6 & 30 & \\ \hline & 2 & 10& -5 & \\ \end{array} \][/tex]
- Add the new result ([tex]\(30\)[/tex]) to the next coefficient ([tex]\(-35\)[/tex]):
[tex]\[-35 + 30 = -5\][/tex]
- Multiply the new result ([tex]\(-5\)[/tex]) by [tex]\(3\)[/tex] and place it under the next coefficient ([tex]\(15\)[/tex]):
[tex]\[-5 \times 3 = -15\][/tex]
[tex]\[ \begin{array}{c|cccc} 3 & 2 & 4 & -35 & 15 \\ \hline & 2 & 6 & 30 & -15 \\ \hline & 2 & 10& -5 & 0 \\ \end{array} \][/tex]
- Add this result ([tex]\(-15\)[/tex]) to the last coefficient ([tex]\(15\)[/tex]):
[tex]\[15 + (-15) = 0\][/tex]
7. Result of synthetic division:
The quotient is given by the numbers we have in the last row except the last term, and the remainder is the last term in the last row.
- Quotient: [tex]\[ 2x^2 + 10x - 5 \][/tex]
- Remainder: [tex]\[ 0 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{2x^2 + 10x - 5} \][/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.
