Using synthetic division find the remainder when
f(x) =2x^4+12x^3+6x^2÷5x+75 is divided by x+5?
You can use this handy calculator to verify your work or see all the details.
https://www.mathportal.org/calculators/polynomials-solvers/synthetic-division-calculator.php
To find the remainder using synthetic division, follow these steps:
Step 1: Write the polynomial expression in descending order of the exponent.
f(x) = 2x^4 + 12x^3 + 6x^2 ÷ 5x + 75
Step 2: Set up the synthetic division table. Place the divisor, x + 5, on the outside of the table and write the coefficients of the dividend, f(x), inside the table, leaving a blank space for the remainder.
-5 | 2 12 6 0 75
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Step 3: Bring down the first coefficient, which is 2, into the bottom row of the synthetic division table.
-5 | 2 12 6 0 75
-10
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Step 4: Multiply the divisor, -5, by the result in the bottom row (which is -10 in this case), and write the product in the next column.
-5 | 2 12 6 0 75
-10
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50
Step 5: Add the product, 50, to the next coefficient in the top row, which is 12. Write the sum in the next column.
-5 | 2 12 6 0 75
-10 20
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50
0
Step 6: Repeat steps 4 and 5 until all coefficients have been processed.
-5 | 2 12 6 0 75
-10 20 -50
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50 20 -44 -220
0
Step 7: The last number in the bottom row, which is -220, is the remainder.
Therefore, when f(x) = 2x^4 + 12x^3 + 6x^2 is divided by x + 5, the remainder is -220.