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Chapter 2: Problem 31

Use synthetic division to divide. $$\left(4 x^{3}-9 x+8 x^{2}-18\right) \div(x+2)$$

Short Answer

Expert verified
The result of the division \( (4x^{3} + 8x^{2} - 9x -18) \div (x+2) \) is \(4x^2 - 9x\).

Step by step solution

01

Set Up Synthetic Division

Create the Synthetic Division grid. Write the coefficients of the original polynomial, \(4, 8, -9, -18\), in the top row. In the lower-left of the grid, place the negative of the numerical value of the divisor, which is -2 in this case.
02

Implement Synthetic Division

Start with bringing down the first coefficient, 4, to the bottom row. Multiply this first number in the bottom row by the number in the lower-left corner (-2), put the result (which is -8) under the second number in the top row and add to find the second number in the bottom row (which will be 0). Repeat the process to complete the synthetic division operation: 0 * -2 = 0, 0 + (-9) = -9, and finally -9 * -2 = 18, then 18+(-18) = 0.
03

Write Down the Answer

The numbers left in the bottom row, 4, 0, -9, 0, represent the coefficients of the answer. Because the original polynomial was a third degree and you divided by a linear term, the result is a second degree polynomial. The answer is \(4x^2 - 9x + 0\) or simply \(4x^2 - 9x\).

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