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Chapter 3: Problem 13

Use long division to divide. Divisor \(x^{2}-1\) Dividend $$x^{3}-27$$

Short Answer

Expert verified
The final answer then becomes \(x\) with the remainder \(-27\). This means \(x^{3}-27\) divided by \(x^{2}-1\) equals \(x\) with a remainder of \(-27\).

Step by step solution

01

Setting up

Set up the long division, putting the dividend \(x^{3}-27\) under the division symbol and the divisor \(x^{2}-1\) to the left.
02

First Division

To start with, divide the leading term of the dividend \(x^{3}\) by the leading term of the divisor \(x^{2}\) to get \(x\). This becomes the first term of the quotient.
03

Multiply and Subtract

Multiply the divisor by the first term of the quotient and subtract this from the dividend. This gives us \(-(x^{3} - (x \cdot x^{2} - x)) = -27 \) as the result.
04

Second Division

Next, divide the leading term of the result by the leading term of the divisor again. This time, since the result is \(-27\) and the divisor is \(x^{2} - 1\), which are incommensurable, there are no more terms to add to the quotient.
05

Write down the remaining terms

As there are no further terms to add, take the remaining terms as the rest. The rest is \(-27\) which makes it the final answer, since we can't subtract anything else from it.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Long Division
Long division is a method used to divide two numbers or expressions, where one is the divisor and the other is the dividend.
For polynomial division, this method involves several repetitive steps of division, multiplication, and subtraction, very much like long division with numbers. It helps to break down complex polynomials into simpler terms or find out how many times a polynomial fits into another.
  • When applied to polynomials, it helps identify a quotient and potentially a remainder.
  • The process continues until all terms being divided are handled.
The systematic nature of long division makes it a powerful tool in algebra, allowing us to simplify expressions that might seem complex at first glance.
Quotient
The quotient is the result obtained from dividing one polynomial by another using long division. In our exercise:
  • The first step is taking the leading term from the dividend and dividing it by the leading term of the divisor.
  • This result gives the first term of the quotient.
Continuing the steps of division, multiplication, and subtraction eventually yields a complete polynomial as the quotient, or it shows that the division resulted in a remainder.
Divisor
The divisor is the polynomial that divides the dividend. It is the expression outside the long division symbol.
In our specific exercise, the divisor is \(x^2 - 1\). It plays a crucial role as dividing begins by comparing the highest degree term of the divisor and the dividend:
  • The degrees of these terms guide the initial division step.
  • The structure of the divisor can indicate possible factors or zeros of the dividend when division is complete.
The divisor directly influences what terms will appear in the quotient, as it's repeatedly multiplied by emerging terms in the quotient.
Dividend
In a long division setup, the dividend is the polynomial placed under the division symbol. It is the expression being divided by the divisor. In the exercise at hand, the dividend is \(x^3 - 27\). Typically:
  • The division begins by focusing on the leading (highest degree) term of the dividend.
  • The rest of the terms are addressed step by step as we progress through the polynomial long division.
Through each step, the dividend is reduced by an amount calculated using the divisor and the emerging terms in the quotient until it possibly becomes zero, or a remainder is left.

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Most popular questions from this chapter

Regression Problem Let \(x\) be the angle (in degrees) at which a baseball is hit with no spin at an initial speed of 40 meters per second and let \(d(x)\) be the distance (in meters) the ball travels. The table shows the distances for the different angles at which the ball is hit. (Source: The Physics of Sports) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|} \hline \text { Angle, } x & 10 & 15 & 30 & 36 & 42 \\ \hline \text { Distance, } d(x) & 58.3 & 79.7 & 126.9 & 136.6 & 140.6 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|l|} \hline \text { Angle, } x & 44 & 45 & 48 & 54 & 60 \\ \hline \text { Distance, } d(x) & 140.9 & 140.9 & 139.3 & 132.5 & 120.5 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. (b) Use the regression feature of a graphing utility to find a quadratic model for \(d(x)\). (c) Use a graphing utility to graph your model for \(d(x)\) with the scatter plot of the data. (d) Find the vertex of the graph of the model from part (c). Interpret its meaning in the context of the problem.

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=2 x^{3}+x^{2}-14 x-10, \quad k=1+\sqrt{3}$$

Credit Cards The numbers of active American Express cards \(C\) (in millions) in the years 1997 to 2006 are shown in the table. (Sourze: American Express) $$ \begin{aligned} &\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 1997 & 1998 & 1999 & 2000 & 2001 \\ \hline \text { Cards, C } & 42.7 & 42.7 & 46.0 & 51.7 & 55.2 \\ \hline \end{array}\\\ &\begin{array}{|l|l|l|l|l|l|} \hline \text { Year } & 2002 & 2003 & 2004 & 2005 & 2006 \\ \hline \text { Cards, C } & 57.3 & 60.5 & 65.4 & 71.0 & 78.0 \\ \hline \end{array} \end{aligned} $$ (a) Use a graphing utility to create a scatter plot of the data. Let \(t\) represent the year, with \(t=7\) corresponding to \(1997 .\) (b) Use what you know about end behavior and the scatter plot from part (a) to predict the sign of the leading coefficient of a quartic model for \(C\). (c) Use the regression feature of a graphing utility to find a quartic model for \(C\). Does your model agree with your answer from part (b)? (d) Use a graphing utility to graph the model from part (c). Use the graph to predict the year in which the number of active American Express cards would be about 92 million. Is your prediction reasonable?

Use long division to divide. Divisor \(x-4\) Dividend $$5 x^{2}-17 x-12$$

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