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Chapter 2: Problem 28

Use synthetic division to divide. $$\left(5 x^{3}+18 x^{2}+7 x-6\right) \div(x+3)$$

Short Answer

Expert verified
The result of the division \((5x^3+18x^2+7x-6) \div(x+3)\) by synthetic division is the polynomial \(5x^2 + 3x + 16\)

Step by step solution

01

Rewrite the Divisor

Recall that synthetic division requires the divisor to take the form \(x - k\). In this case, the divisor is \(x + 3\), which we can rewrite as \(x - (-3)\) to meet the requirement. Contact -3 for synthetic division is chosen.
02

Set up the Synthetic Division Table

To set up the synthetic division table, write down the coefficients of the polynomial \(5x^3 + 18x^2 + 7x - 6\), which are 5, 18, 7, and -6. Write these numbers in a row. Place the number -3 (chosen from Step 1) in a 'box' to the left of this row. Draw a horizontal line below the row.
03

Perform Synthetic Division

Start the operation by bringing down the first number (5). Multiply the number in the box (-3) by this number and write the result (-15) below the next number in the row (18). Add these two numbers together (18 - 15) and write the result (3) below the line. Repeat this process: multiply the number in the box by the new number below the line and write this result below the next number in the row. The result for this step looks like this:\[\begin{array}{c|cccc}-3 & 5 & 18 & 7 & -6 \& \downarrow & -15 & 9 & -6 \\hline& 5 & 3 & 16 & 0\end{array}\]
04

Interpret the Result

The numbers below the line now represent the coefficients of the polynomial that is the result of the division. The degrees of the polynomial powers descend by one from the original polynomial, starting from degree 2 (since we divided by a linear polynomial): \(5x^2 + 3x + 16\). Since the last number is zero, there is no remainder.

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