Use synthetic division to determine the remainder when 5x4 + 7x3 + x2 − 18 is divided by x + 2.
Question
Use synthetic division to determine the remainder when is divided by .
Solution
Sure, here's how you can use synthetic division to find the remainder when 5x^4 + 7x^3 + x^2 - 18 is divided by x + 2.
Step 1: Write down the coefficients of the dividend (5x^4 + 7x^3 + x^2 - 18), which are 5, 7, 1, 0 (for x), and -18.
Step 2: Write down the zero of the divisor (x + 2). The zero is the solution to the equation x + 2 = 0, which is x = -2.
Step 3: Set up the synthetic division. Draw a line, write -2 on the left, and the coefficients from step 1 on the right:
-2 | 5 7 1 0 -18
Step 4: Bring down the first coefficient (5) to start the division:
-2 | 5 7 1 0 -18
-----------------
5
Step 5: Multiply the number you just wrote (5) by -2 (the zero of the divisor), and write the result (-10) under the next coefficient (7):
-2 | 5 7 1 0 -18
- 10
-----------------
5 -3
Step 6: Add the numbers in the second column (7 and -10), and write the result (-3) under the line:
-2 | 5 7 1 0 -18
- 10
-----------------
5 -3
Step 7: Repeat steps 5 and 6 for the rest of the coefficients. Multiply -3 by -2 to get 6, write it under 1, add to get 7. Multiply 7 by -2 to get -14, write it under 0, add to get -14. Multiply -14 by -2 to get 28, write it under -18, add to get 10.
-2 | 5 7 1 0 -18
- 10 6 -14 28
-----------------
5 -3 7 -14 10
Step 8: The numbers under the line represent the coefficients of the quotient and the remainder. The last number (10) is the remainder. So, the remainder when 5x^4 + 7x^3 + x^2 - 18 is divided by x + 2 is 10.
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