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Chapter 2: Problem 38

Solve each polynomial inequality and graph the solution set on a real number line. Express each solution set in interval notation. $$x^{3}+7 x^{2}-x-7<0$$

Short Answer

Expert verified
The solution set of the given inequality is (-鈭�, -7) 鈭� (1, 鈭�).

Step by step solution

01

Factorize the Polynomial

Rearrange the given polynomial inequality \(x^{3}+7 x^{2}-x-7<0\) to the standard polynomial form: \(x^{3}+7x^{2}-x-7=0\). The roots of this polynomial will be the critical points dividing the number line. Use the Rational Root Theorem to factorize the polynomial. The rational roots are x = -7, x = 1, and x = 1.
02

Plot the Roots on the Number Line

Plot the roots determined in Step 1 on the number line. This divides the number line into four intervals(-鈭�, -7), (-7, 1), (1, 鈭�). Remember, these are open intervals, so they do not include the endpoint values.
03

Determine the Sign of the Polynomial in Each Interval

Choose a test point within each interval, and substitute it into the polynomial to determine the sign. For interval (-鈭�, -7), choose -8, for interval (-7, 1), choose 0, and for interval (1, 鈭�), choose 2. Substituting these values into the polynomial, it is seen that the polynomial is negative in the intervals (-鈭�, -7) and (1, 鈭�) and positive in the interval (-7, 1).
04

Express the Solution in Interval Notation

Since we are looking for intervals where the polynomial is less than zero, the solution will be (-鈭�, -7) 鈭� (1, 鈭�).

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Most popular questions from this chapter

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 of \(f(x)=\frac{1}{x}\) to graph \(g.\) $$g(x)=\frac{3 x+7}{x+2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The inequality \(\frac{x-2}{x+3}<2\) can be solved by multiplying both sides by \(x+3,\) resulting in the equivalent inequality \(x-2<2(x+3)\).

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a. If \(y=\frac{k}{x},\) find the value of \(k\) using \(x=8\) and \(y=12\). b. Substitute the value for \(k\) into \(y=\frac{k}{x}\) and write the resulting equation. c. Use the equation from part (b) to find \(y\) when \(x=3\).
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