To divide the polynomial [tex]\( 28x^5 + 29x^4 + 5x^3 + 86x^2 + 56x + 53 \)[/tex] by [tex]\(-4x - 7\)[/tex] using synthetic division, let's walk through the steps:
### Step-by-Step Solution
1. Identify the coefficients of the polynomial:
The coefficients of the polynomial [tex]\( 28x^5 + 29x^4 + 5x^3 + 86x^2 + 56x + 53 \)[/tex] in order are:
[tex]\[ 28, 29, 5, 86, 56, 53 \][/tex]
2. Adjust the divisor:
The divisor is [tex]\(-4x - 7\)[/tex]. For synthetic division, we use the root of the divisor's linear factor. Solve for [tex]\( x \)[/tex]:
[tex]\[ -4x - 7 = 0 \implies x = -\frac{7}{-4} = \frac{7}{4} \][/tex]
3. Set up the synthetic division:
The process starts by bringing down the leading coefficient (the first coefficient of the polynomial), which is 28.
4. Perform synthetic division:
- Multiply 28 by the root [tex]\(\frac{7}{4}\)[/tex] and add the result to the next coefficient (29):
[tex]\[ 28 \times \frac{7}{4} = 49 \][/tex]
[tex]\[ 29 + 49 = 78 \][/tex]
- Multiply 78 by [tex]\(\frac{7}{4}\)[/tex] and add the result to the next coefficient (5):
[tex]\[ 78 \times \frac{7}{4} = 136.5 \][/tex]
[tex]\[ 5 + 136.5 = 141.5 \][/tex]
- Multiply 141.5 by [tex]\(\frac{7}{4}\)[/tex] and add the result to the next coefficient (86):
[tex]\[ 141.5 \times \frac{7}{4} = 247.125 \][/tex]
[tex]\[ 86 + 247.125 = 333.625 \][/tex]
- Multiply 333.625 by [tex]\(\frac{7}{4}\)[/tex] and add the result to the next coefficient (56):
[tex]\[ 333.625 \times \frac{7}{4} = 583.84375 \][/tex]
[tex]\[ 56 + 583.84375 = 639.84375 \][/tex]
- Multiply 639.84375 by [tex]\(\frac{7}{4}\)[/tex] and add the result to the next coefficient (53):
[tex]\[ 639.84375 \times \frac{7}{4} = 1119.7265625 \][/tex]
[tex]\[ 53 + 1119.7265625 = 1172.7265625 \][/tex]
### The Result
After performing synthetic division, we find:
- The coefficients of the quotient polynomial are:
[tex]\[ 28, 78.0, 141.5, 333.625, 639.84375 \][/tex]
- The remainder is:
[tex]\[ 1172.7265625 \][/tex]
Thus, the polynomial division results in the quotient polynomial:
[tex]\[ 28x^4 + 78x^3 + 141.5x^2 + 333.625x + 639.84375 \][/tex]
And the remainder is:
[tex]\[ 1172.7265625 \][/tex]
