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Chapter 3: Problem 57

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 division of \(2 x^{3}-3 x^{2}-11x+7\) by \(x-3\) is \(2x^2 + 3x - 8 + \frac{1}{x-3}\).

Step by step solution

01

Initiate the Division

The division process of \(2 x^{3}-3 x^{2}-11x+7\) by \(x-3\) is commenced by dividing the first term (leading term) of the numerator by the first term of the denominator, which gives \(2x^2\).
02

Multiply and Subtract

Next, \(x-3\) is multiplied by the result obtained in step 1, \(2x^2\), yielding \(2x^{3} - 6x^{2}\). Then, subtract this from the original numerator \(2 x^{3}-3 x^{2}-11x+7\), which gives the new numerator \(3x^2 - 11x + 7\).
03

Repeat the Process

Now, the process should be repeated by dividing the leading term of the updated numerator by the leading term of the denominator, yielding the next term of the quotient, 3x. Then, the denominator \(x-3\) should be multiplied by the newly obtained term 3x, and the result is subtracted from the updated numerator. This results in the new numerator \( - 8x + 7\).
04

Finish the Long Division

Repeat the same operation: divide -8x by x to obtain -8, and multiply \(x-3\) by -8. Then subtract this from the final numerator to get a remainder of 1. Thus, the result of the division, or the quotient is \(2x^2 + 3x -8\) with the remainder 1.
05

Append the Remainder

Include the remainder as a fraction over the divisor to the end of the quotient, so the full representation of the solution becomes \(2x^2 + 3x - 8 + \frac{1}{x-3}\).

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Most popular questions from this chapter

Use the position function $$ s(t)--16 t^{2}+v_{0} t+s_{0} $$ \(\left(v_{11}=\text { initial velocity, } s_{0}-\text { initial position, } t-\text { time }\right)\) to answer Exercises \(75-76\) You throw a ball straight up from a rooftop 160 feet high with an initial velocity of 48 feet per second. During which time period will the ball's height exceed that of the rooftop?

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{1}{x+1}>\frac{2}{x-1} $$

Use long division to rewrite the equation for \(g\) in the form $$\text {quotient }+\frac{\text {remainder}}{\text {divisor}}$$ Then use this form of the function's equation and transformations. $$g(x)=\frac{3 x+7}{x+2}$$

Is every rational function a polynomial function? Why or why not? Does a true statement result if the two adjectives rational and polynomial are reversed? Explain.

Solve each inequality in Exercises \(65-70\) and graph the solution set on a real number line. $$ \frac{3}{x+3}>\frac{3}{x-2} $$
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