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Chapter 2: Problem 34

Use synthetic division to divide. $$\left(3 x^{3}-16 x^{2}-72\right) \div(x-6)$$

Short Answer

Expert verified
The result of the synthetic division \(3x^3 - 16x^2 - 72\) divided by \(x-6\) is \(3x^2+2x+12\).

Step by step solution

01

Set up Synthetic Division

First, put the coefficients of the polynomial functions in the correct order. The dividend is \(3x^3 - 16x^2 - 72\) and the divisor is \(x-6\). Write down the coefficients \(3, -16, 0, -72\) (note that there is a hidden 0 coefficient for \(x^1\) in the dividend). Below these coefficients, write the root of the divisor, that is \(6\) from \(x-6\). The setup should look like this: \n\n\n 6 | 3 -16 0 -72\n ----------------------
02

Synthetic Division

Now, carry down the first coefficient \(3\) directly below the line and multiply it with the root (\(6\)) and put the result under the second column (In this case -16). Subtract from \(-16\). This value is then multiplied by the root again and put under the third column. Continue this process until you have gone through all the terms. The division should look like this: \n\n\n 6 | 3 -16 0 -72\n | 18 12 72\n ------------------------ \n | 3 2 12 0
03

Write the Result

Finally, write down the result of the synthetic division as a polynomial. The coefficients \(3, 2, 12, 0\) represent the polynomial \(3x^2+2x+12\) plus no remainder.

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