Answer :
To perform the division [tex]\(x - 1 \longdiv x^2 + 4x + 5\)[/tex] using synthetic division, follow these steps:
1. Set up the synthetic division:
- The divisor is [tex]\(x - 1\)[/tex]. In synthetic division, we use the root of the divisor, which is [tex]\(1\)[/tex].
- Write down the coefficients of the dividend [tex]\(x^2 + 4x + 5\)[/tex], which are [tex]\(1\)[/tex], [tex]\(4\)[/tex], and [tex]\(5\)[/tex].
2. Arrange the synthetic division:
- Write down the number [tex]\(1\)[/tex] (root of the divisor) on the left.
- Write the coefficients of the dividend in a row: [tex]\(1, 4, 5\)[/tex].
3. Perform the synthetic division:
- Step 1: Bring down the first coefficient.
- The first coefficient is [tex]\(1\)[/tex]; write it below the line.
- Step 2: Multiply the number you just wrote ([tex]\(1\)[/tex]) by the root ([tex]\(1\)[/tex]) and write the result under the next coefficient.
- [tex]\(1 \times 1 = 1\)[/tex]. Write [tex]\(1\)[/tex] under the next coefficient [tex]\(4\)[/tex].
- Step 3: Add the result to the next coefficient.
- [tex]\(4 + 1 = 5\)[/tex]. Write [tex]\(5\)[/tex] below the line.
- Step 4: Repeat the process: multiply the latest result by the root, and then add it to the next coefficient.
- Multiply [tex]\(5\)[/tex] by [tex]\(1\)[/tex], which is [tex]\(5\)[/tex], and add it to the last coefficient [tex]\(5\)[/tex].
- [tex]\(5 + 5 = 10\)[/tex]. Write [tex]\(10\)[/tex] below the line.
4. Write the result:
- The numbers below the line represent the coefficients of the quotient and the remainder.
- The quotient is [tex]\(x + 5\)[/tex] (from the coefficients [tex]\(1\)[/tex] and [tex]\(5\)[/tex]).
- The remainder is [tex]\(10\)[/tex].
Therefore, the synthetic division of [tex]\(x^2 + 4x + 5\)[/tex] by [tex]\(x - 1\)[/tex] results in a quotient of [tex]\(x + 5\)[/tex] with a remainder of [tex]\(10\)[/tex]. The numbers that fill the blanks in the original problem statement based on the synthetic layout are [tex]\(1\)[/tex] for the dividend, representing the coefficients [tex]\(1, 4, 5\)[/tex] in the division setup.
1. Set up the synthetic division:
- The divisor is [tex]\(x - 1\)[/tex]. In synthetic division, we use the root of the divisor, which is [tex]\(1\)[/tex].
- Write down the coefficients of the dividend [tex]\(x^2 + 4x + 5\)[/tex], which are [tex]\(1\)[/tex], [tex]\(4\)[/tex], and [tex]\(5\)[/tex].
2. Arrange the synthetic division:
- Write down the number [tex]\(1\)[/tex] (root of the divisor) on the left.
- Write the coefficients of the dividend in a row: [tex]\(1, 4, 5\)[/tex].
3. Perform the synthetic division:
- Step 1: Bring down the first coefficient.
- The first coefficient is [tex]\(1\)[/tex]; write it below the line.
- Step 2: Multiply the number you just wrote ([tex]\(1\)[/tex]) by the root ([tex]\(1\)[/tex]) and write the result under the next coefficient.
- [tex]\(1 \times 1 = 1\)[/tex]. Write [tex]\(1\)[/tex] under the next coefficient [tex]\(4\)[/tex].
- Step 3: Add the result to the next coefficient.
- [tex]\(4 + 1 = 5\)[/tex]. Write [tex]\(5\)[/tex] below the line.
- Step 4: Repeat the process: multiply the latest result by the root, and then add it to the next coefficient.
- Multiply [tex]\(5\)[/tex] by [tex]\(1\)[/tex], which is [tex]\(5\)[/tex], and add it to the last coefficient [tex]\(5\)[/tex].
- [tex]\(5 + 5 = 10\)[/tex]. Write [tex]\(10\)[/tex] below the line.
4. Write the result:
- The numbers below the line represent the coefficients of the quotient and the remainder.
- The quotient is [tex]\(x + 5\)[/tex] (from the coefficients [tex]\(1\)[/tex] and [tex]\(5\)[/tex]).
- The remainder is [tex]\(10\)[/tex].
Therefore, the synthetic division of [tex]\(x^2 + 4x + 5\)[/tex] by [tex]\(x - 1\)[/tex] results in a quotient of [tex]\(x + 5\)[/tex] with a remainder of [tex]\(10\)[/tex]. The numbers that fill the blanks in the original problem statement based on the synthetic layout are [tex]\(1\)[/tex] for the dividend, representing the coefficients [tex]\(1, 4, 5\)[/tex] in the division setup.
