Sure, let's go through the synthetic division step-by-step to find the quotient and remainder when [tex]\( x^4 - 6x^3 + 3x^2 + 10x + 2 \)[/tex] is divided by [tex]\( x - 5 \)[/tex].
### Step-by-Step Solution
The polynomial [tex]\( x^4 - 6x^3 + 3x^2 + 10x + 2 \)[/tex] is divided by [tex]\( x - 5 \)[/tex]. In synthetic division, we use the root of the divisor, which is 5 in this case.
(a) Fill out the synthetic division table:
1. Write down the coefficients of the polynomial: 1, -6, 3, 10, 2.
2. Bring the first coefficient down: 1.
Now, we follow these steps iteratively:
- Multiply this first coefficient by 5 (the root of the divisor).
- Write the result under the next coefficient.
- Add the numbers in the column and write the result below the line.
Let's fill the table step by step.
1 | [tex]\( \begin{array}{c|ccccc} 5 & 1 & -6 & 3 & 10 & 2 \\ \hline & 1 & & & & \end{array} \)[/tex]
First Iteration:
- Multiply the current result (1) by 5: [tex]\( 1 \times 5 = 5 \)[/tex].
- Add this to the next coefficient: [tex]\( -6 + 5 = -1 \)[/tex].
1 | [tex]\( \begin{array}{c|ccccc} 5 & 1 & -6 & 3 & 10 & 2 \\ \hline & 1 & -1 & & & \end{array} \)[/tex]
Second Iteration:
- Multiply the current result (-1) by 5: [tex]\( -1 \times 5 = -5 \)[/tex].
- Add this to the next coefficient: [tex]\( 3 + (-5) = -2 \)[/tex].
1 | [tex]\( \begin{array}{c|ccccc} 5 & 1 & -6 & 3 & 10 & 2 \\ \hline & 1 & -1 & -2 & & \end{array} \)[/tex]
Third Iteration:
- Multiply the current result (-2) by 5: [tex]\( -2 \times 5 = -10 \)[/tex].
- Add this to the next coefficient: [tex]\( 10 + (-10) = 0 \)[/tex].
1 | [tex]\( \begin{array}{c|ccccc} 5 & 1 & -6 & 3 & 10 & 2 \\ \hline & 1 & -1 & -2 & 0 & \end{array} \)[/tex]
Fourth Iteration:
- Multiply the current result (0) by 5: [tex]\( 0 \times 5 = 0 \)[/tex].
- Add this to the next coefficient: [tex]\( 2 + 0 = 2 \)[/tex].
1 | [tex]\( \begin{array}{c|ccccc} 5 & 1 & -6 & 3 & 10 & 2 \\ \hline & 1 & -1 & -2 & 0 & 2 \end{array} \)[/tex]
The synthetic division table is now completed:
[tex]\[ 1, -1, -2, 0, 2 \][/tex]
(b) Write your answer in the form of Quotient [tex]\(\frac{\text{Remainder}}{x - 5}\)[/tex]:
The quotient is given by the results except for the last number: [tex]\( 1, -1, -2, 0 \)[/tex]. The last number, 2, is the remainder.
So, the quotient in polynomial form is [tex]\( x^3 - x^2 - 2x + 0 \)[/tex] (or simply [tex]\( x^3 - x^2 - 2x \)[/tex]). The remainder is 2.
Therefore, the division can be written as:
[tex]\[ \frac{x^4 - 6x^3 + 3x^2 + 10x + 2}{x - 5} = x^3 - x^2 - 2x + \frac{2}{x - 5} \][/tex]
So our final answer is:
[tex]\[ x^3 - x^2 - 2x + \frac{2}{x - 5} \][/tex]
