To divide the polynomial [tex]\( 28x^5 + 29x^4 + 5x^3 + 86x^2 + 56x + 53 \)[/tex] by [tex]\( -4x - 7 \)[/tex] using synthetic division, follow these steps:
### Step 1: Adjust the Divisor
First, we rewrite the divisor [tex]\( -4x - 7 \)[/tex] as [tex]\( 4x + 7 \)[/tex] since synthetic division method assumes a monic (leading coefficient of 1) divisor. This means we will work with the zero of the divisor [tex]\( \frac{-7}{4} \)[/tex] or [tex]\( -\frac{7}{-4} = \frac{7}{4} \)[/tex].
### Step 2: Setup the Synthetic Division
We use the coefficients of the polynomial:
[tex]\[ \text{Dividend: } [28, 29, 5, 86, 56, 53] \][/tex]
And the divisor:
[tex]\[ \text{Divisor: } \frac{7}{4} \][/tex]
### Step 3: Perform Synthetic Division
1. Initialize:
Write down the leading coefficient of the dividend (28) in the first row.
[tex]\[
\begin{array}{r|rrrrrr}
\frac{7}{4} & 28 & 29 & 5 & 86 & 56 & 53 \\
\hline
& 28 \\
\end{array}
\][/tex]
2. Multiply and add:
- Multiply the coefficient by the divisor and write it under the next coefficient.
- Sum this with the next coefficient and write the result below the line.
Continue this process for each coefficient:
[tex]\[
\begin{array}{r|rrrrrr}
\frac{7}{4} & 28 & 29 & 5 & 86 & 56 & 53 \\
\hline
& 28 & 78 & 141.5 & 333.625 & 639.84375 & 1172.7265625 \\
\end{array}
\][/tex]
The values under the divider are computed as follows:
- 29 + 28 [tex]\(\cdot \frac{7}{4}\)[/tex] = 78
- 5 + 78 [tex]\(\cdot \frac{7}{4}\)[/tex] = 141.5
- 86 + 141.5 [tex]\(\cdot \frac{7}{4}\)[/tex] = 333.625
- 56 + 333.625 [tex]\(\cdot \frac{7}{4}\)[/tex] = 639.84375
- 53 + 639.84375 [tex]\(\cdot \frac{7}{4}\)[/tex] = 1172.7265625
### Step 4: Write Quotient and Remainder
The coefficients under the division bar except the last one make up the quotient. The last value is the remainder.
So the quotient is:
[tex]\[ 28x^4 + 78x^3 + 141.5x^2 + 333.625x + 639.84375 \][/tex]
And the remainder is:
[tex]\[ 1172.7265625 \][/tex]
### Step 5: Combine the Results
Combining these results, we get:
[tex]\[ 28x^5 + 29x^4 + 5x^3 + 86x^2 + 56x + 53 = (28x^4 + 78x^3 + 141.5x^2 + 333.625x + 639.84375)(-4x - 7) + 1172.7265625 \][/tex]
Expressing the remainder in the form involving [tex]\( -4x - 7 \)[/tex]:
[tex]\[ \frac{1172.7265625}{-4x - 7} \][/tex]
Finally, the complete quotient and remainder representation is:
[tex]\[ \boxed{28x^4 + 78x^3 + 141.5x^2 + 333.625x + 639.84375 + \frac{1172.7265625}{-4x - 7}} \][/tex]
