Answer :
To divide the polynomial [tex]\(19x^3 + 0x^2 + 16x - 128\)[/tex] by [tex]\(x - 4\)[/tex] using synthetic division, follow these steps:
1. Set up the synthetic division process:
- Write down the coefficients of the polynomial: [tex]\(19, 0, 16, -128\)[/tex].
- The divisor is [tex]\(x - 4\)[/tex], so we use [tex]\(4\)[/tex] in the synthetic division process.
2. Perform the synthetic division:
- Bring down the first coefficient, [tex]\(19\)[/tex], to start.
- Multiply [tex]\(19\)[/tex] by [tex]\(4\)[/tex] (the divisor root) and write the result under the next coefficient. Add this to the next coefficient [tex]\(0\)[/tex].
- [tex]\(19 \times 4 = 76\)[/tex]
- [tex]\(0 + 76 = 76\)[/tex]
- Write [tex]\(76\)[/tex] as the next number in the row below.
- Repeat the process: Multiply [tex]\(76\)[/tex] by [tex]\(4\)[/tex] and add to the next coefficient, [tex]\(16\)[/tex].
- [tex]\(76 \times 4 = 304\)[/tex]
- [tex]\(16 + 304 = 320\)[/tex]
- Write [tex]\(320\)[/tex] as the next number.
- Again, multiply [tex]\(320\)[/tex] by [tex]\(4\)[/tex] and add to the next coefficient [tex]\(-128\)[/tex].
- [tex]\(320 \times 4 = 1280\)[/tex]
- [tex]\(-128 + 1280 = 1152\)[/tex]
3. Interpret the results:
- The numbers at the bottom row, except the last one, represent the coefficients of the quotient polynomial: [tex]\(19x^2 + 76x + 320\)[/tex].
- The last number is the remainder: [tex]\(1152\)[/tex].
Therefore, when you divide [tex]\(19x^3 + 0x^2 + 16x - 128\)[/tex] by [tex]\(x - 4\)[/tex], you get the quotient [tex]\(19x^2 + 76x + 320\)[/tex] with a remainder of [tex]\(1152\)[/tex].
1. Set up the synthetic division process:
- Write down the coefficients of the polynomial: [tex]\(19, 0, 16, -128\)[/tex].
- The divisor is [tex]\(x - 4\)[/tex], so we use [tex]\(4\)[/tex] in the synthetic division process.
2. Perform the synthetic division:
- Bring down the first coefficient, [tex]\(19\)[/tex], to start.
- Multiply [tex]\(19\)[/tex] by [tex]\(4\)[/tex] (the divisor root) and write the result under the next coefficient. Add this to the next coefficient [tex]\(0\)[/tex].
- [tex]\(19 \times 4 = 76\)[/tex]
- [tex]\(0 + 76 = 76\)[/tex]
- Write [tex]\(76\)[/tex] as the next number in the row below.
- Repeat the process: Multiply [tex]\(76\)[/tex] by [tex]\(4\)[/tex] and add to the next coefficient, [tex]\(16\)[/tex].
- [tex]\(76 \times 4 = 304\)[/tex]
- [tex]\(16 + 304 = 320\)[/tex]
- Write [tex]\(320\)[/tex] as the next number.
- Again, multiply [tex]\(320\)[/tex] by [tex]\(4\)[/tex] and add to the next coefficient [tex]\(-128\)[/tex].
- [tex]\(320 \times 4 = 1280\)[/tex]
- [tex]\(-128 + 1280 = 1152\)[/tex]
3. Interpret the results:
- The numbers at the bottom row, except the last one, represent the coefficients of the quotient polynomial: [tex]\(19x^2 + 76x + 320\)[/tex].
- The last number is the remainder: [tex]\(1152\)[/tex].
Therefore, when you divide [tex]\(19x^3 + 0x^2 + 16x - 128\)[/tex] by [tex]\(x - 4\)[/tex], you get the quotient [tex]\(19x^2 + 76x + 320\)[/tex] with a remainder of [tex]\(1152\)[/tex].
