Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 25
Capitolo 4, Problema 25

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x2 ≤ 4x − 2

Guida verificata passo dopo passo
1
Rewrite the inequality \(x^2 \leq 4x - 2\) by bringing all terms to one side to set the inequality to zero: \(x^2 - 4x + 2 \leq 0\).
Identify the quadratic expression \(x^2 - 4x + 2\) and find its roots by solving the equation \(x^2 - 4x + 2 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=1\), \(b=-4\), and \(c=2\).
Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots and then find the exact roots using the quadratic formula.
Use the roots to divide the real number line into intervals. Test a value from each interval in the inequality \(x^2 - 4x + 2 \leq 0\) to determine where the inequality holds true.
Express the solution set as an interval or union of intervals based on the test results, and graph this solution set on the real number line.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Polynomial Inequalities

Polynomial inequalities involve expressions where a polynomial is compared to another value using inequality symbols (e.g., ≤, ≥, <, >). Solving them requires finding the values of the variable that make the inequality true, often by analyzing the sign of the polynomial over different intervals.
Video consigliato:
06:07
Linear Inequalities

Factoring and Finding Critical Points

To solve polynomial inequalities, rewrite the inequality so one side is zero, then factor the polynomial if possible. The roots or zeros of the polynomial, called critical points, divide the number line into intervals where the polynomial's sign can be tested to determine where the inequality holds.
Video consigliato:
04:36
Factor by Grouping

Interval Notation and Graphing Solution Sets

After determining the intervals where the inequality is true, express the solution set using interval notation, which concisely describes all values satisfying the inequality. Graphing on a real number line visually represents these intervals, showing included endpoints with closed dots for ≤ or ≥.
Video consigliato:
05:18
Interval Notation
Pratica correlata
Domanda del libro di testo

In Exercises 25–26, graph each polynomial function. f(x)=2x2(x−1)3(x+2)f(x) = 2x^2(x - 1)^3(x + 2)

990
views
Domanda del libro di testo

Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=4−(x−1)2

1101
views
Domanda del libro di testo

Find an nth-degree polynomial function with real coefficients satisfying the given conditions. If you are using a graphing utility, use it to graph the function and verify the real zeros and the given function value. n=3; 1 and 5i are zeros; f(-1) = -104

760
views
Domanda del libro di testo

Divide using synthetic division. (x2−5x−5x3+x4)÷(5+x)

504
views
Domanda del libro di testo

Find the zeros for each polynomial function and give the multiplicity for each zero. State whether the graph crosses the x-axis, or touches the x-axis and turns around, at each zero. f(x)=2(x−5)(x+4)2

1270
views
Domanda del libro di testo

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=x/x(x+4)

642
views