what is the remainder of the synthetic division problem below
What is the remainder of the synthetic division problem below?
To determine the remainder of a synthetic division problem, we follow a specific process. Let’s illustrate this with an example, assuming we want to divide a polynomial P(x) by a binomial of the form x - c.
Steps for Synthetic Division
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Write down the coefficients of the polynomial: For example, if we have P(x) = 2x^3 - 3x^2 + 4x - 5, we would write down the coefficients as ( [2, -3, 4, -5] ).
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Determine the value of c in x - c: If we are dividing by x - 3, then c = 3.
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Set up the synthetic division table:
- Write the value of c on the left.
- Write the coefficients of P(x) on the top row.
Here’s how it looks:
3 | 2 -3 4 -5
- Carry down the leading coefficient (the first number in the row of coefficients):
- In this example, it’s 2.
3 | 2 -3 4 -5
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2
- Multiply c by the value just written below the line and write the result under the next coefficient:
- 3 \times 2 = 6.
3 | 2 -3 4 -5
----------------
2 6
- Add the values in this column and write the result beneath the line:
- -3 + 6 = 3.
3 | 2 -3 4 -5
----------------
2 3
- Repeat the process:
- Multiply c by the value just obtained (3) and write the result under the next coefficient.
- 3 \times 3 = 9.
3 | 2 -3 4 -5
----------------
2 3 9
- Add the values in this column: 4 + 9 = 13.
3 | 2 -3 4 -5
----------------
2 3 13
- Multiply c by the value just obtained (13) and write the result under the next coefficient.
- 3 \times 13 = 39.
3 | 2 -3 4 -5
----------------
2 3 13 39
- Add the values in this column: -5 + 39 = 34.
3 | 2 -3 4 -5
----------------
2 3 13 34
The last value at the bottom is the remainder.
Answer:
In this specific example of dividing ( 2x^3 - 3x^2 + 4x - 5 ) by ( x - 3 ), the remainder is 34.
If you provide the specific polynomial and divisor for your problem, I can apply the same steps to find the remainder.