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Chapter 3: Problem 76

Find the quotient of \(x^{3 n}+1\) and \(x^{n}+1\)

Short Answer

Expert verified
The quotient of \(x^{3 n}+1\) and \(x^{n}+1\) is \(x^{2 n}\).

Step by step solution

01

Identify Terms for Division

We need to find the quotient when \(x^{3 n}+1\) is divided by \(x^n+1\). Here, \(x^{3 n}+1\) is the dividend and \(x^{n}+1\) is the divisor.
02

Perform Division using Polynomial Division

Now, we perform the division as we would in regular polynomial division.\n Divide \(x^{3 n}\) by \(x^{n}\) to get \(x^{2 n}\). Then multiply \(x^{n}+1\) by the result, which gives \( x^{3 n} + x^{2 n}\). Subtract this from \(x^{3 n}+1\) to get \(- x^{2 n} + 1\). As the degree of \(- x^{2 n} + 1\) is less than the degree of \(x^{n}+1\), we stop here, and \(x^{2 n}\) is our quotient.
03

Result

The quotient of the division \(x^{3 n}+1\) divided by \(x^{n}+1\) is \(x^{2 n}\).

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Most popular questions from this chapter

Find the axis of symmetry for each parabola whose equation is given. Use the axis of symmetry to find a second point on the parabola whose \(y\) -coordinate is the same as the given point. $$f(x)=(x-3)^{2}+2 ; \quad(6,11)$$

Make Sense? In Exercises \(94-97\), determine whether each statement makes sense or does not make sense, and explain your reasoning. When solving \(f(x)>0,\) where \(f\) is a polynomial function, 1 only pay attention to the sign of \(f\) at each test value and not the actual function value.

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. $$g(x)=\frac{2 x+7}{x+3}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of a rational function can have three vertical asymptotes.

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.
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