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Chapter 2: Problem 29

Use synthetic division to divide. \(\left(6 x^{3}+7 x^{2}-x+26\right) \div(x-3)\)

Short Answer

Expert verified
The division \( \left(6 x^{3}+7 x^{2}-x+26\right) \div(x-3) \) results in \(6x^{2}+25x-76 - \frac{202}{x-3}\)

Step by step solution

01

Set Up the Synthetic Division

To set up the synthetic division, we write the coefficients of the polynomial \(\left(6, 7, -1, 26\right)\) in a row. We then write the zero of the divisor (x - 3 has a zero at x = 3) to the left of the coefficient bar.
02

Carry Out the Synthetic Division

Here's how to carry it out: 1. Drop down the first coefficient \((6)\). 2. Multiply the zero of the divisor \((3)\) by the number \(6\) and write the result below the next coefficient \(7\).3. Add 7 and 18 to get 25 and write it below the bar.4. Multiply again the obtained \(25\) by the divisor's zero \(3\) and write the result \(-75\) below \(-1\).5. Add \(-1\) and \(-75\) to get \(-76\).6. Multiply \(-76\) by \(3\) to get \(-228\).7. Add \(-228\) and \(26\) to get \(-202\) and write this below the bar.
03

Interpret the Results

The answer isn't just \(-202\), but each number on the bottom row. The first number (6), seeing that we divided a cubic by a linear function, is the coefficient of the \(x^2\) term in the answer. The next number (25) is the coefficient of the linear \(x\) term in the answer. The last number (\(-202\)) is the constant of our division. Thus, our final division result is \(6x^2 + 25x - 76\) with a remainder of \(-202\). So, the division can be written as: \(6x^{2}+25x-76 - \frac{202}{x-3}\)

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