Use synthetic division to find p(3) for p(x)=x^4-2x^3-4x+4
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To find p(3) using synthetic division, follow these steps:
Step 1: Write down the coefficients of the polynomial in descending order of exponents:
p(x) = x^4 - 2x^3 - 4x + 4
The coefficients are 1, -2, -4, 0, and 4.
Step 2: Set up the synthetic division table, where the divisor is 3:
3 | 1 -2 -4 0 4
Step 3: Bring down the first coefficient:
3 | 1 -2 -4 0 4
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Step 4: Multiply the divisor (3) by the previous result (1), and write the product below the next coefficient:
3 | 1 -2 -4 0 4
3
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Step 5: Add the product (3) to the next coefficient (-2), and write the sum below:
3 | 1 -2 -4 0 4
3
------------------
1
Step 6: Repeat Steps 4 and 5 for the remaining coefficients:
3 | 1 -2 -4 0 4
3 3
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1 1
3 | 1 -2 -4 0 4
3 3 9
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1 1 5
3 | 1 -2 -4 0 4
3 3 9 9
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1 1 5 9
3 | 1 -2 -4 0 4
3 3 9 9 27
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1 1 5 9 31
Step 7: The remainder is the last entry in the synthetic division table, which is 31.
Step 8: Therefore, p(3) = 31.
To find p(3) using synthetic division, follow these steps:
Step 1: Write the polynomial in descending order with missing terms represented by placeholders:
p(x) = x^4 - 2x^3 + 0x^2 - 4x + 4
Step 2: Set up the synthetic division table by writing the constant term (-4) outside the division symbol and the coefficients of the polynomial (1 -2 0 -4 4) inside the division symbol.
3 | 1 -2 0 -4 4
Step 3: Bring down the first coefficient, which is 1, directly below the division symbol:
3 | 1 -2 0 -4 4
------------------------
|
Step 4: Multiply the number outside the division symbol (3) by the number just brought down (1) and write the result below the next coefficient (-2):
3 | 1 -2 0 -4 4
------------------------
| 3
Step 5: Add the result you just obtained (3) to the next coefficient (-2) and write it below the line:
3 | 1 -2 0 -4 4
------------------------
| 3
|__________
| 1 1
Step 6: Repeat steps 4 and 5, continuing with the next coefficient and the result obtained:
3 | 1 -2 0 -4 4
------------------------
| 3
|__________
| 1 1
| 3 [ 12]
Step 7: Continue this process until you have performed it for all the coefficients. You will end up with a final row of numbers.
3 | 1 -2 0 -4 4
------------------------
| 3
|__________
| 1 1
| 3 [ 12]
| 4 [ 12]
Step 8: The last number on the bottom row is the remainder. In this case, the remainder is 12.
Step 9: The other numbers on the bottom row, from left to right, are the coefficients of the quotient polynomial. For this polynomial, they are 1, 1, 3, and 4.
Step 10: Write the quotient polynomial using the coefficients obtained:
Quotient polynomial = 1x^3 + 1x^2 + 3x + 4
Step 11: The value of p(3) is equal to the remainder obtained, which is 12.
Therefore, p(3) = 12.
try this site - it shows all the steps
http://www.mathportal.org/calculators/polynomials-solvers/synthetic-division-calculator.php