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Chapter 2: Problem 37

Use synthetic division to divide. $$\frac{10 x^{4}-50 x^{3}-800}{x-6}$$

Short Answer

Expert verified
\(10x^3 - 50x^2 - 300x -1800 - \frac{64800}{x - 6}\)

Step by step solution

01

Setup the Synthetic Division

To setup synthetic division, write down the coefficients of the polynomial, including the coefficients for any missing terms and set up a division-like format on the side. The divisor is \(x - 6\) , so the number used in synthetic division will be 6. Write it outside and to the left of the synthetic division bar, like so: \[6 | 10, -50, 0, 0, -800\] Note that we have included 0's for the missing terms \(x^2\) and \(x\).
02

Perform the Synthetic Division

Bring down the leading coefficient (10), multiply by 6 (the number used), and place this value under the next coefficient (-50). Add these values and repeat the process: multiply this sum by 6, and place this under the next coefficient (0). Continue this process until all coefficients have been used. The work will appear as follows: \[6 | 10, -50, -300, -1800, -10800, 64800 \] The last number (64800) is the remainder.
03

Write the Result

Lastly, we need to write the result of the synthetic division as a polynomial plus any remainder as a fraction over the divisor. Each number is associated with a term (with degree decreasing from left to right) of the result. The result of the synthetic division is: \(10x^3 - 50x^2 - 300x -1800 - \frac{64800}{x - 6}\)

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