Domanda del libro di testo
Among all pairs of numbers whose difference is 14, find a pair whose product is as small as possible. What is the minimum product?
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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 9
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. x2−6x+9<0
Guida verificata passo dopo passo1
Recognize that the inequality is a quadratic inequality: \(x^2 - 6x + 9 < 0\).
Factor the quadratic expression on the left side. Notice that \(x^2 - 6x + 9\) is a perfect square trinomial, so it factors as \((x - 3)^2\).
Rewrite the inequality using the factorization: \((x - 3)^2 < 0\).
Consider the properties of squares: since \((x - 3)^2\) is always greater than or equal to zero for all real \(x\), it can never be less than zero.
Conclude that there are no real values of \(x\) that satisfy \((x - 3)^2 < 0\), so the solution set is the empty set, which in interval notation is \(\emptyset\).
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Polynomial Inequalities
Polynomial inequalities involve expressions where a polynomial is compared to zero or another value using inequality symbols like <, >, ≤, or ≥. 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 Quadratic Expressions
Factoring is the process of rewriting a quadratic expression as a product of two binomials. For example, x² - 6x + 9 factors to (x - 3)(x - 3). Factoring helps identify the roots of the polynomial, which are critical points for determining where the inequality changes sign.
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06:08Solving Quadratic Equations by Factoring
Interval Notation and Number Line Graphing
Interval notation is a concise way to represent sets of real numbers, especially solution sets of inequalities. Graphing on a number line visually shows where the polynomial is positive or negative. Together, they help communicate the solution clearly, indicating which intervals satisfy the inequality.
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05:18Interval Notation
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In Exercises 9–16, a) List all possible rational zeros. b) Use synthetic division to test the possible rational zeros and find an actual zero. c) Use the quotient from part (b) to find the remaining zeros of the polynomial function. f(x)=x3+x2−4x−4
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Domanda del libro di testo
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Domanda del libro di testo
Use the four-step procedure for solving variation problems given on page 447 to solve Exercises 1–10. y varies jointly as x and z. y = 25 when x = 2 and z = 5. Find y when x = 8 and z = 12.
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Domanda del libro di testo
Find the domain of each rational function. f(x)=(x+7)/(x2+49)
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Domanda del libro di testo
In Exercises 1–10, determine which functions are polynomial functions. For those that are, identify the degree. f(x)=x1/3 −4x2+7
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