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Chapter 3: Problem 19

Divide using synthetic division. $$\left(3 x^{2}+7 x-20\right) \div(x+5)$$

Short Answer

Expert verified
The result of the division \((3 x^{2}+7 x-20) \div(x+5)\) using synthetic division is \(3x - 8 + \frac{20}{x+5}\)

Step by step solution

01

Setup for Synthetic Division

To set up for synthetic division, write the coefficients of the dividing polynomial (here, \(3 x^{2}+7 x-20\) ) in order and arrange them horizontally. To do this, write down the coefficients of each term in descending order of power. Thus, the coefficients would be 3, 7, and -20. Below and to the left of them, write the number which makes the divisor zero. In this case, it is -5 since that is what renders \(x+5\) equal to zero.
02

Perform the Synthetic Division

Begin by bringing down the leading coefficient (which is 3 in this case) to the bottom row. Afterwards, multiply this number by -5 (the number you wrote to the left), and write the result (i.e., -15) under the second coefficient (i.e., 7) in the top row, then add straight down to get -8. Repeat this process: Multiply -8 by -5 to get 40 and write this under -20, then add to get 20.
03

Interpret the Result

The numbers in the bottom row represent the coefficients of the polynomial result of the division. The final result chronicles from the highest power (one less than the power of the original polynomial) to the lowest. Consequently, the result of the synthetic division is \(3x - 8 + \frac{20}{x+5}\). Please note that the final term, 20, is the remainder and is divided by the original divisor, \(x+5\).

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Most popular questions from this chapter

In Exercises \(35-50\) a. Use the Leading Coefficient Test to determine the graphs end behavior. b. Find \(x\) -intercepts by setting \(f(x)=0\) and solving the resulting polynomial equation. State whether the graph crosses the \(x\)-axis, or touches the \(x\)-axis and turns around, at each intercept. c. Find the \(y\) -intercept by setting \(x\) equal to 0 and computing \(f(0)\) d. Determine whether the graph has \(y\) -axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the fact that the maximum number of turning points of the graph is \(n-1\) to check whether it is drawn correctly. $$f(x)=6 x-x^{3}-x^{5}$$

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