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Chapter 2: Problem 3

Write the domain of the given function \(r\) as a union of intervals. $$ r(x)=\frac{4 x^{7}+8 x^{2}-1}{x^{2}-2 x-6} $$

Short Answer

Expert verified
The domain of the function \(r(x)\) is expressed as a union of intervals: \(\boxed{(-\infty,1-\sqrt{7})\cup(1-\sqrt{7},1+\sqrt{7})\cup(1+\sqrt{7},\infty)}\).

Step by step solution

01

Identify the denominator of the rational function

In the given function, \( r(x)=\frac{4 x^{7}+8 x^{2}-1}{x^{2}-2 x-6} \) The denominator is the part that is under the division sign, which is \(x^{2}-2 x-6\).
02

Find the values of x that make the denominator zero

Next, we will set the denominator equal to zero and solve for x: \(x^{2}-2 x-6=0\) To solve this, we can look for factors, or use the quadratic formula: \( x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a} \) For this equation, \(a = 1\), \(b = -2\), and \(c = -6\). Plug these values into the formula: \( x=\frac{2\pm\sqrt{(-2)^{2}-4(1)(-6)}}{2(1)} \)
03

Calculate the values of x that make the denominator zero

Now, we will simplify and solve for x: \( x=\frac{2\pm\sqrt{4+24}}{2} \) \( x=\frac{2\pm\sqrt{28}}{2} \) We can simplify \(\sqrt{28}\) as follows: \( \sqrt{28}=\sqrt{4\times 7}=2\sqrt{7} \) Now, we have two values for x: \( x=\frac{2+2\sqrt{7}}{2} \text{ or } x=\frac{2-2\sqrt{7}}{2} \) \( x=1+\sqrt{7} \text{ or } x=1-\sqrt{7} \)
04

Express the domain as a union of intervals

For the function to be defined, the denominator must not be equal to zero. We found that the denominator is equal to zero when \(x=1+\sqrt{7}\) or \(x=1-\sqrt{7}\), so we will exclude these two values from the domain. The domain of the function, expressed as a union of intervals, will be: \(\boxed{(-\infty,1-\sqrt{7})\cup(1-\sqrt{7},1+\sqrt{7})\cup(1+\sqrt{7},\infty)}\)

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Most popular questions from this chapter

Find a polynomial \(p\) of degree 3 such that \(-2,-1,\) and 4 are zeros of \(p\) and \(p(1)=2\).

Find a number \(c\) such that -2 is a zero of the polynomial \(p\) defined by $$ p(x)=5-3 x+4 x^{2}+c x^{3} $$.

Write the indicated expression as a ratio of polynomials, assuming that $$ r(x)=\frac{3 x+4}{x^{2}+1}, \quad s(x)=\frac{x^{2}+2}{2 x-1}, \quad t(x)=\frac{5}{4 x^{3}+3} $$. $$ (s \circ r)(x) $$

Write each expression as the sum of a polynomial and a rational function whose numerator has smaller degree than its denominator. $$ \frac{2 x+1}{x-3} $$

Write the indicated expression as \(a\) polynomial. $$ \frac{s(1+x)-s(1)}{x} $$
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