Textbook Question
Use the quotient rule to simplify the expressions in Exercises 23–32. Assume that x > 0. √48x3/√3x
Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
Not the one you use?Change textbookSelect a different textbook
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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
6mWas this helpful?
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.
Recommended video:
Guided course
6:29
Factor Using Special Product Formulas
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.
Recommended video:
Guided course
05:20Expanding Radicals
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.
Recommended video:
06:36Solving Quadratic Equations Using The Quadratic Formula
Related Practice
Textbook Question
In Exercises 15–32, multiply or divide as indicated. (x2+x)/(x2−4) ÷ (x2−1)/(x2+5x+6)
4
views
Textbook Question
In Exercises 15–32, multiply or divide as indicated. (x2−4)/(x−2) ÷ (x+2)/(4x−8)
1
views
Textbook Question
Evaluate each exponential expression: (33)/(36)
9
views
Textbook Question
Find the intersection of the sets. {a,b,c,d}∩∅
5
views
Textbook Question
Find each product. (2x−5)(7x+2)
1
views