교과서 질문
Find all complex zeros of each polynomial function. Give exact values. List multiple zeros as necessary.* ƒ(x)=x4+2x3-3x2+24x-180
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Ch. 3 - Polynomial and Rational Functions
Lial13th EditionCollege AlgebraISBN: 9780136881063
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Table of contents
4장, 문제 95
Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. -x2 + 4x + 1 ≥ 0
검증된 단계별 안내1
Rewrite the inequality in a standard form by moving all terms to one side: \(-x^{2} + 4x + 1 \geq 0\).
Multiply the entire inequality by \(-1\) to make the quadratic coefficient positive, remembering to reverse the inequality sign: \(x^{2} - 4x - 1 \leq 0\).
Find the roots of the quadratic equation \(x^{2} - 4x - 1 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=1\), \(b=-4\), and \(c=-1\).
Determine the intervals defined by the roots and test values within each interval to see where the inequality \(x^{2} - 4x - 1 \leq 0\) holds true.
Write the solution set in interval notation based on the intervals where the inequality is satisfied, including endpoints if the inequality is non-strict (\(\geq\) or \(\leq\)).
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주요 개념
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Solving Quadratic Inequalities
Quadratic inequalities involve expressions where a quadratic polynomial is compared to zero or another value using inequality signs. To solve them, first rewrite the inequality in standard form, then find the roots of the corresponding quadratic equation. These roots divide the number line into intervals to test for where the inequality holds true.
추천 영상:
04:03
Choosing a Method to Solve Quadratics
Factoring and Finding Roots of Quadratic Equations
Finding the roots of a quadratic equation involves setting the quadratic expression equal to zero and solving for the variable. This can be done by factoring, completing the square, or using the quadratic formula. The roots are critical points that help determine the intervals for testing the inequality.
추천 영상:
06:08Solving Quadratic Equations by Factoring
Interval Notation and Solution Sets
Interval notation is a concise way to represent sets of real numbers that satisfy inequalities. It uses parentheses and brackets to indicate open or closed intervals, respectively. After determining where the quadratic inequality holds, express the solution set using interval notation to clearly communicate the range of valid values.
추천 영상:
05:18Interval Notation
관련 실천
교과서 질문
The following exercises are geometric in nature and lead to polynomial models. Solve each problem. A standard piece of notebook paper measuring 8.5 in. by 11 in. is to be made into a box with an open top by cutting equal-size squares from each cor-ner and folding up the sides. Let x represent the length of a side of each such square in inches. Use the table feature of a graphing calculator to do the following. Round to the nearest hundredth.
a. Find the maximum volume of the box.
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교과서 질문
Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. -x2 + 2x + 6 > 0
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교과서 질문
Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function.
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교과서 질문
The following exercises are geometric in nature and lead to polynomial models. Solve each problem. A certain right triangle has area 84 in.2. One leg of the triangle measures 1 in. less than the hypotenuse. Let x represent the length of the hypotenuse. Express the length of the leg mentioned above in terms of x. Give the domain of x.
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교과서 질문
Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. 3x2 + x ≥ 4
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