Use synthetic division to find P(3) for P(x) = x^4 - 6x^3 - 4x^2 - 6x - 2
I don't know how to solve this, can someone explain how please?
I thank you for the explanation, not so much the sarcasm. Reason I didn't just divide was because my teacher had said it was more than just division. But thanks for the help! ♥
To find P(3) for the given polynomial P(x) = x^4 - 6x^3 - 4x^2 - 6x - 2 using synthetic division, follow these steps:
Step 1: Set up the synthetic division table:
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3 | 1 -6 -4 -6 -2
Step 2: Write down the coefficients of the polynomial in descending order in the first row of the table. In this case, the coefficients are 1, -6, -4, -6, and -2.
Step 3: The number outside the table represents the value at which you want to evaluate the polynomial, in this case, 3.
Step 4: Bring down the first coefficient (1) into the second row of the table. This is always the same for synthetic division.
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3 | 1
Step 5: Multiply the number outside the table (3) by the number in the second row of the table (1). Write the result in the third row.
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3 | 1
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3
Step 6: Add the value in the third row to the second coefficient (-6). Write the result in the fourth row.
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3 | 1 -6
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3
Step 7: Multiply the number outside the table (3) by the number in the fourth row (3). Write the result in the fifth row.
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3 | 1 -6
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3 3
Step 8: Add the value in the fifth row to the third coefficient (-4). Write the result in the sixth row.
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3 | 1 -6 -4
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3 3
Step 9: Multiply the number outside the table (3) by the number in the sixth row (3). Write the result in the seventh row.
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3 | 1 -6 -4
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3 3 9
Step 10: Add the value in the seventh row to the fourth coefficient (-6). Write the result in the eighth row.
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3 | 1 -6 -4 -6
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3 3 9
Step 11: Multiply the number outside the table (3) by the number in the eighth row (9). Write the result in the ninth row.
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3 | 1 -6 -4 -6
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3 3 9 27
Step 12: Add the value in the ninth row to the fifth coefficient (-2). Write the result in the tenth row.
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3 | 1 -6 -4 -6 -2
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3 3 9 27 25
Step 13: The value in the tenth row is the remainder when P(3) is divided by (x - 3). In this case, the remainder is 25.
Step 14: To find the value of P(3), add the remainder to the constant term (-2). P(3) = 25 - 2 = 23.
Therefore, P(3) = 23.
To evaluate P(3) for the given polynomial P(x), you can use synthetic division. Synthetic division is a method used to divide a polynomial by a linear binomial of the form (x - c), where c is a constant.
1. Set up the synthetic division table by writing down the coefficients of the polynomial in descending order. Include placeholders for any missing terms.
- For the given polynomial P(x) = x^4 - 6x^3 - 4x^2 - 6x - 2, the coefficients are:
1 | -6 | -4 | -6 | -2
- The first number, 1, represents the coefficient of the highest degree term x^4. The last number, -2, represents the constant term.
2. Write the value, 3, of P(x) that you want to evaluate outside the table.
3. Carry out the synthetic division by dividing each coefficient in the table by the divisor (3) and following these steps:
- Bring down the first coefficient, which is 1, directly below the division line.
- Multiply this number (1) by the divisor (3) and write the result below the next coefficient (-6).
- Add these two numbers (-18) and write the sum below the next coefficient (-4).
- Repeat this process until you reach the last coefficient:
1 | -6 | -4 | -6 | -2
-18 | -66 | -204
- The final number, -204, is the remainder.
4. The number in the bottom row of the synthetic division table is the value of P(3).
- In this case, P(3) = -204.
Therefore, P(3) for the given polynomial is -204.
I can not imagine why you would divide.
3^4 - 6 * 3^3 - 4 * 3^2 - 6 * 3 - 2
= 81 - 162 - 36 - 18 - 2