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Chapter 3: Problem 20

Divide using synthetic division. $$\left(5 x^{2}-12 x-8\right) \div(x+3)$$

Short Answer

Expert verified
The result of the division \(5x^{2} - 12x -8\) divided by \(x +3\) is \( 5x - 27 + \frac{73}{x+3}\)

Step by step solution

01

Set Up the Synthetic Division Tableau

Write the coefficients of the polynomial \(5x^{2}-12x-8\) in descending order of their respective terms' degree, which gives 5, -12, -8. Draw a horizontal line under these numbers. To the left of the line, write the number that makes the divisor \(x+3\) equal to zero. To find this, set \(x+3\) equal to zero and solve for \(x\), giving \(x=-3\). Write -3 to the left of the horizontal line.
02

Fill the Tableau

Drop the first number (5) straight down below the line. Then multiply -3 by 5, write the result (-15) under -12 and add to get -27. Multiply -3 by -27, write the result (81) below -8, add to get 73.
03

Interpret the Result

The numbers below the line represent the coefficients of the quotient polynomial. The degree of the polynomial decreases by one. This means the result of the division is \(5x - 27\). The remainder is 73, so the full answer to the division is \(5x - 27 + \frac{73}{x+3}\).

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