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Chapter 3: Problem 47

Explain how to perform long division of polynomials. Use \(2 x^{3}-3 x^{2}-11 x+7\) divided by \(x-3\) in your explanation.

Short Answer

Expert verified
The result of the long division of \(2x^{3} - 3x^{2} - 11x + 7\) by \(x-3\) is \(2x^{2} + 3x + 2 + \frac{1}{x-3}\).

Step by step solution

01

Divide

First, divide the leading term in the dividend, \(2x^{3}\), by the leading term in the divisor, \(x\). This gives you \(2x^{2}\), the first term of the quotient.
02

Multiply

Next, multiply the divisor \(x-3\) by the first term of the quotient \(2x^{2}\). This gives \(2x^{3} - 6x^{2}\). Write this under the dividend aligned by like terms.
03

Subtract

Subtract the result from the previous step (\(2x^{3} - 6x^{2}\)) from the first two terms of the dividend. This involves subtracting each term individually. Then, we get \((-3x^{2}) - (-6x^{2}) = 3x^{2}\).
04

Bring Down

To continue the division, bring down the next term of the dividend, which is \(-11x\). Now the new dividend for the next round of division is \(3x^{2} - 11x\). Repeat steps 1-4 until you've brought down all terms.
05

Continue Division

You will continue with the divisions until the degree of the remainder is less than the degree of the original divisor. The remaining expression is the remainder. Thus, your polynomial fraction can be expressed as the quotient plus the remainder over the divisor.

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Most popular questions from this chapter

In Exercises \(1-10\), determine which functions are polynomial functions. For those that are, identify the degree. $$f(x)=7 x^{2}+9 x^{4}$$

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. $$f(x)=x^{4}-6 x^{3}+9 x^{2}$$

Explain how to use the Leading Coefficient Test to determine the end behavior of a polynomial function.

Use a graphing utility to obtain a complete graph for each polynomial function in Exercises \(58-61 .\) Then determine the number of real zeros and the number of nonreal complex zeros for each function. $$ f(x)=x^{6}-64 $$

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