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Answer :
To perform the division [tex]\(x - 1\)[/tex] into [tex]\(x^2 + 4x + 5\)[/tex] using synthetic division, we follow these steps:
1. Identify the divisor and dividend:
- The divisor is [tex]\(x - 1\)[/tex], which means we use the number 1 in our synthetic division.
- The dividend is [tex]\(x^2 + 4x + 5\)[/tex], so the coefficients are [tex]\([1, 4, 5]\)[/tex].
2. Set up the synthetic division:
- Write 1 on the left side, which comes from the divisor [tex]\(x - 1\)[/tex] (we take the opposite sign of the constant in the divisor).
- Write the coefficients of the dividend: 1, 4, and 5.
3. Begin the synthetic division process:
- Bring down the first coefficient (1) to the bottom row. This is the starting value.
4. Perform the synthetic division across the coefficients:
- Multiply the number just written (1) by the number we used from the divisor (1), and write the result under the next coefficient (4). So, [tex]\(1 \times 1 = 1\)[/tex].
- Add this result to the next coefficient: [tex]\(4 + 1 = 5\)[/tex]. Write this sum below the line.
- Repeat this process: multiply the last result (5) by 1, which gives 5, and add this to the next coefficient (5). So, [tex]\(5 + 5 = 10\)[/tex].
5. Extract the results:
- The numbers just calculated give us the coefficients of the quotient polynomial.
- The last number is the remainder. In this case, the quotient is [tex]\([1, 3]\)[/tex] and the remainder is [tex]\(2\)[/tex].
Therefore, the quotient of the division is [tex]\(x + 3\)[/tex] and the remainder is 2. So, the division [tex]\(x^2 + 4x + 5\)[/tex] by [tex]\(x - 1\)[/tex] can be expressed as:
[tex]\[ x^2 + 4x + 5 = (x - 1)(x + 3) + 2 \][/tex]
1. Identify the divisor and dividend:
- The divisor is [tex]\(x - 1\)[/tex], which means we use the number 1 in our synthetic division.
- The dividend is [tex]\(x^2 + 4x + 5\)[/tex], so the coefficients are [tex]\([1, 4, 5]\)[/tex].
2. Set up the synthetic division:
- Write 1 on the left side, which comes from the divisor [tex]\(x - 1\)[/tex] (we take the opposite sign of the constant in the divisor).
- Write the coefficients of the dividend: 1, 4, and 5.
3. Begin the synthetic division process:
- Bring down the first coefficient (1) to the bottom row. This is the starting value.
4. Perform the synthetic division across the coefficients:
- Multiply the number just written (1) by the number we used from the divisor (1), and write the result under the next coefficient (4). So, [tex]\(1 \times 1 = 1\)[/tex].
- Add this result to the next coefficient: [tex]\(4 + 1 = 5\)[/tex]. Write this sum below the line.
- Repeat this process: multiply the last result (5) by 1, which gives 5, and add this to the next coefficient (5). So, [tex]\(5 + 5 = 10\)[/tex].
5. Extract the results:
- The numbers just calculated give us the coefficients of the quotient polynomial.
- The last number is the remainder. In this case, the quotient is [tex]\([1, 3]\)[/tex] and the remainder is [tex]\(2\)[/tex].
Therefore, the quotient of the division is [tex]\(x + 3\)[/tex] and the remainder is 2. So, the division [tex]\(x^2 + 4x + 5\)[/tex] by [tex]\(x - 1\)[/tex] can be expressed as:
[tex]\[ x^2 + 4x + 5 = (x - 1)(x + 3) + 2 \][/tex]
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