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Chapter 3: Problem 22

Divide using synthetic division. $$\left(5 x^{3}-6 x^{2}+3 x+11\right) \div(x-2)$$

Short Answer

Expert verified
The result of the division is \(5x^2 - 16x + 32 + \frac{75}{x-2}\).

Step by step solution

01

Set Up the Synthetic Division

Write the coefficients of the polynomial to be divided at the top of the synthetic division table. For the polynomial \(5x^3 - 6x^2 + 3x + 11\), the coefficients are 5, -6, 3, and 11. Then, to the left of the bar, write the value that makes the divisor equal to zero; in this case, the expression is \(x - 2\), so the value is 2.
02

Execute the Synthetic Division

Drop the first coefficient (5) down to the bottom row. Multiply this coefficient by the value from the divisor (2), then add the product to the next coefficient in the top row (-6) to get the next coefficient for the bottom row. Repeat this process until all coefficients have been used.
03

Interpret the Results

The bottom row of coefficients represent the coefficients of the quotient. The degree of the quotient is always one less than the degree of the dividend. The last number in the bottom row is the remainder. Thus, the quotient polynomial is \(5x^2 - 16x + 32\), and the remainder is 75.

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Most popular questions from this chapter

Which one of the following is true? a. If \(f(x)=-x^{3}+4 x,\) then the graph of \(f\) falls to the left and to the right. b. A mathematical model that is a polynomial of degree \(n\) whose leading term is \(a_{n} x^{n}, n\) odd and \(a_{n}<0,\) is ideally suited to describe nonnegative phenomena over unlimited periods of time. c. There is more than one third-degree polynomial function with the same three \(x\) -intercepts. d. The graph of a function with origin symmetry can rise to the left and to the right.

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The rational function $$f(x)=\frac{27,725(x-14)}{x^{2}+9}-5 x$$ models the number of arrests, \(f(x)\), per \(100,000\) drivers, for driving under the influence of alcohol, as a function of a driver's age, \(x\) a. Graph the function in a \([0,70,5]\) by \([0,400,20]\) viewing rectangle. b. Describe the trend shown by the graph. c. Use the ZOOM and TRACE features or the maximum function feature of your graphing utility to find the age that corresponds to the greatest number of arrests. How many arrests, per \(100,000\) drivers, are there for this age group?
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