IDNLearn.com is your go-to resource for finding expert answers and community support. Our community provides accurate and timely answers to help you understand and solve any issue.
Sagot :
Sure, let's perform synthetic division step by step to divide the polynomial [tex]\(4x^3 + 8x^2 - 4x + 9\)[/tex] by [tex]\(x + 3\)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients of the polynomial:
The polynomial [tex]\(4x^3 + 8x^2 - 4x + 9\)[/tex] has coefficients 4, 8, -4, and 9.
2. Find the zero of the divisor:
The divisor is [tex]\(x + 3\)[/tex], which means [tex]\(x = -3\)[/tex] is the zero of the divisor.
3. Set up the synthetic division:
We will write the coefficients in a row: [tex]\(4, 8, -4, 9\)[/tex].
We write the zero of the divisor, which is [tex]\(-3\)[/tex], to the left.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ \end{array} \][/tex]
4. Perform the synthetic division steps:
- Start by bringing down the first coefficient (4) directly below the line.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & & & \\ \end{array} \][/tex]
- Multiply the zero [tex]\(-3\)[/tex] by the number just written below the line, and write the result below the next coefficient (8).
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -12 & & \\ \end{array} \][/tex]
- Add this result to the next coefficient (8), and write the sum below the line.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & & \\ \end{array} \][/tex]
- Repeat the process: multiply [tex]\(-3\)[/tex] by the latest number written below the line ([tex]\(-4\)[/tex]), and write the result below the next coefficient ([tex]\(-4\)[/tex]).
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & 12 & \\ \end{array} \][/tex]
- Add this to the next coefficient ([tex]\(-4\)[/tex]), and write the sum below the line.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & 8 & \\ \end{array} \][/tex]
- Repeat the process one more time: multiply [tex]\(-3\)[/tex] by the latest number written below the line (8), and write the result below the next coefficient (9).
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & 8 & -24 \\ \end{array} \][/tex]
- Add this to the next coefficient (9), and write the sum below the line.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & 8 & -15 \\ \end{array} \][/tex]
5. Identify the quotient and the remainder:
- The numbers below the line, excluding the very last one, give us the coefficients of the quotient polynomial.
- The last number is the remainder.
So, the quotient is [tex]\(4x^2 - 4x + 8\)[/tex] and the remainder is [tex]\(-15\)[/tex].
Therefore, the final answers are:
- The quotient is [tex]\(Q(x) = 4x^2 - 4x + 8\)[/tex].
- The remainder is [tex]\(R(x) = -15\)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients of the polynomial:
The polynomial [tex]\(4x^3 + 8x^2 - 4x + 9\)[/tex] has coefficients 4, 8, -4, and 9.
2. Find the zero of the divisor:
The divisor is [tex]\(x + 3\)[/tex], which means [tex]\(x = -3\)[/tex] is the zero of the divisor.
3. Set up the synthetic division:
We will write the coefficients in a row: [tex]\(4, 8, -4, 9\)[/tex].
We write the zero of the divisor, which is [tex]\(-3\)[/tex], to the left.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ \end{array} \][/tex]
4. Perform the synthetic division steps:
- Start by bringing down the first coefficient (4) directly below the line.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & & & \\ \end{array} \][/tex]
- Multiply the zero [tex]\(-3\)[/tex] by the number just written below the line, and write the result below the next coefficient (8).
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -12 & & \\ \end{array} \][/tex]
- Add this result to the next coefficient (8), and write the sum below the line.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & & \\ \end{array} \][/tex]
- Repeat the process: multiply [tex]\(-3\)[/tex] by the latest number written below the line ([tex]\(-4\)[/tex]), and write the result below the next coefficient ([tex]\(-4\)[/tex]).
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & 12 & \\ \end{array} \][/tex]
- Add this to the next coefficient ([tex]\(-4\)[/tex]), and write the sum below the line.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & 8 & \\ \end{array} \][/tex]
- Repeat the process one more time: multiply [tex]\(-3\)[/tex] by the latest number written below the line (8), and write the result below the next coefficient (9).
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & 8 & -24 \\ \end{array} \][/tex]
- Add this to the next coefficient (9), and write the sum below the line.
[tex]\[ \begin{array}{r|rrrr} -3 & 4 & 8 & -4 & 9 \\ & & 4 & -4 & 8 & -15 \\ \end{array} \][/tex]
5. Identify the quotient and the remainder:
- The numbers below the line, excluding the very last one, give us the coefficients of the quotient polynomial.
- The last number is the remainder.
So, the quotient is [tex]\(4x^2 - 4x + 8\)[/tex] and the remainder is [tex]\(-15\)[/tex].
Therefore, the final answers are:
- The quotient is [tex]\(Q(x) = 4x^2 - 4x + 8\)[/tex].
- The remainder is [tex]\(R(x) = -15\)[/tex].
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. IDNLearn.com is committed to providing accurate answers. Thanks for stopping by, and see you next time for more solutions.
