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Chapter 3: Problem 23

Divide using synthetic division. $$\left(6 x^{5}-2 x^{3}+4 x^{2}-3 x+1\right) \div(x-2)$$

Short Answer

Expert verified
The solution is \(6x^4 + 10x^3 + 20x^2 + 40x + 77 + \frac{155}{x-2}\)

Step by step solution

01

Arrange the synthetic division set up

First, the coefficients from the polynomial \(6 x^{5}-2 x^{3}+4 x^{2}-3 x+1\) are arranged over a synthetic division bar. The number from the divisor \(x-2\) (which is 2) is placed outside the L-shape bar. Include 0s for any missing terms.
02

Synthetic Division Process

Bring down the first coefficient (6) which lands under the solution bar. Multiply this by 2 (the number outside the L-shape) to get 12. Add this to the next coefficient (-2) to get 10, which also places under the solution bar. Repeat this process until you reach the last coefficient.
03

Express the result as a polynomial

The numbers under the solution bar become the coefficients of the answer polynomial. As we've used a 5th degree polynomial, the answer polynomial starts with degree 4 (one less). Write these coefficients with corresponding variables with decreasing powers. The last number under the solution bar is the remainder from division, and should be written as a fraction over the divisor. If there's no remainder, this step can be omitted.

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Most popular questions from this chapter

Use a graphing utility to graph $$f(x)=\frac{x^{2}-4 x+3}{x-2} \quad \text { and} \quad g(x)=\frac{x^{2}-5 x+6}{x-2}$$ What differences do you observe between the graph of \(f\) and \(g ?\) How do you account for these differences?

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. $$f(x)=x^{4}-9 x^{2}$$

In Exercises \(27-34,\) find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the \(x\) -axis, or touches the \(x\) -axis and turns around, at each zero. $$f(x)=2(x-5)(x+4)^{2}$$

Use a graphing utility to graph \(y=\frac{1}{x^{2}}, y=\frac{1}{x^{4}},\) and \(y=\frac{1}{x^{6}}\) in the same viewing rectangle. For even values of \(n,\) how does changing \(n\) affect the graph of \(y=\frac{1}{x^{n}} ?\)

Use everyday language to describe the behavior of a graph near its vertical asymptote if \(f(x) \rightarrow \infty\) as \(x \rightarrow-2^{-}\) and \(f(x) \rightarrow-\infty\) as \(x \rightarrow-2^{+}\).
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