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Divide using synthetic division. (x7+x5−10x3+12)/(x+2)
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 4, Problema 28
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. 4x2−4x+1≥0
Guida verificata passo dopo passo1
Start by recognizing that the inequality is a quadratic inequality: \(4x^2 - 4x + 1 \geq 0\).
Find the roots of the corresponding quadratic equation \(4x^2 - 4x + 1 = 0\) by using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a=4\), \(b=-4\), and \(c=1\).
Calculate the discriminant \(\Delta = b^2 - 4ac\) to determine the nature of the roots.
Use the roots (if any) to divide the real number line into intervals. Test a value from each interval in the original inequality to determine where the inequality holds true.
Express the solution set in interval notation based on the intervals where the inequality is satisfied, and then graph this solution set on the real number line.
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Polynomial Inequalities
Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols (>, <, ≥, ≤). 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.
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06:07
Linear Inequalities
Factoring and Quadratic Expressions
Factoring is the process of rewriting a polynomial as a product of simpler polynomials. For quadratic expressions like 4x²−4x+1, recognizing perfect square trinomials or using the quadratic formula helps identify roots, which are critical points for determining where the polynomial changes sign.
Video consigliato:
06:08Solving Quadratic Equations by Factoring
Interval Notation and Graphing on the Number Line
Interval notation is a concise way to represent sets of real numbers that satisfy inequalities, using parentheses and brackets to indicate open or closed intervals. Graphing the solution on a number line visually shows where the polynomial inequality holds true, aiding in understanding the solution set.
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05:18Interval Notation
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