Textbook Question
Evaluate each exponential expression: 5-3• 5
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- 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
Chapter 1, Problem 27
In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 6x2−11x+4
Verified step by step guidance1
Identify the trinomial: \(6x^2 - 11x + 4\). The goal is to factor it into the form \((ax + b)(cx + d)\), where \(a\), \(b\), \(c\), and \(d\) are constants.
Multiply the leading coefficient (6) and the constant term (4): \(6 \times 4 = 24\). Now, find two numbers that multiply to 24 and add to the middle coefficient, \(-11\). These numbers are \(-8\) and \(-3\).
Rewrite the middle term \(-11x\) as \(-8x - 3x\): \(6x^2 - 8x - 3x + 4\). This step splits the trinomial into four terms to allow factoring by grouping.
Group the terms into two pairs: \((6x^2 - 8x) - (3x - 4)\). Factor out the greatest common factor (GCF) from each group: \(2x(3x - 4) - 1(3x - 4)\).
Notice that \((3x - 4)\) is a common factor. Factor it out: \((2x - 1)(3x - 4)\). This is the factored form of the trinomial.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Trinomials
Factoring trinomials involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. This process requires identifying two numbers that multiply to ac (the product of a and c) and add to b. Understanding this concept is crucial for simplifying expressions and solving equations.
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Prime Trinomials
A prime trinomial is a quadratic expression that cannot be factored into simpler binomial expressions with rational coefficients. Recognizing when a trinomial is prime is essential, as it indicates that the expression cannot be simplified further. This concept helps in determining the nature of the roots of the quadratic equation.
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Quadratic Formula
The quadratic formula, x = (-b ± √(b² - 4ac)) / (2a), provides a method for finding the roots of any quadratic equation. This formula is particularly useful when factoring is difficult or when determining if a trinomial is prime. Understanding how to apply the quadratic formula can help verify the results of factoring.
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Related Practice
Textbook Question
Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √48x3/√3x
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Textbook Question
In Exercises 15–32, multiply or divide as indicated. (x2+x)/(x2−4) ÷ (x2−1)/(x2+5x+6)
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Textbook Question
Simplify each exponential expression:
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Textbook Question
Evaluate each exponential expression: (33)/(36)
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Textbook Question
Find the intersection of the sets. {a,b,c,d}∩∅
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