Textbook Question
Add or subtract terms whenever possible.
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 36
In Exercises 17–38, factor each trinomial, or state that the trinomial is prime. 3x2+4xy+y2
Verified step by step guidance1
Identify the trinomial: \(3x^2 + 4xy + y^2\). This is a quadratic trinomial in terms of \(x\) and \(y\). The goal is to factor it into two binomials, if possible.
Check if the trinomial is in standard form: \(ax^2 + bxy + cy^2\). Here, \(a = 3\), \(b = 4\), and \(c = 1\).
Multiply \(a\) and \(c\): \(3 \times 1 = 3\). Now, find two numbers that multiply to \(3\) and add to \(b = 4\). These numbers are \(3\) and \(1\).
Rewrite the middle term \(4xy\) using the two numbers found: \(3x^2 + 3xy + xy + y^2\). This step splits the middle term to facilitate factoring by grouping.
Group the terms in pairs and factor each group: \((3x^2 + 3xy) + (xy + y^2)\). Factor out the greatest common factor (GCF) from each group: \(3x(x + y) + y(x + y)\). Finally, factor out the common binomial \((x + y)\): \((3x + y)(x + y)\).
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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 the coefficient of x^2 and the constant term) and add to b (the coefficient of x). Understanding this concept is essential for simplifying expressions and solving equations.
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Prime Trinomials
A prime trinomial is a quadratic expression that cannot be factored into the product of two binomials with rational coefficients. Recognizing when a trinomial is prime is crucial, as it indicates that the expression cannot be simplified further. This concept helps in determining the nature of the roots of the quadratic equation associated with the trinomial.
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The Discriminant
The discriminant, given by the formula b^2 - 4ac for a quadratic equation ax^2 + bx + c, provides insight into the nature of the roots of the equation. If the discriminant is positive, there are two distinct real roots; if it is zero, there is one real root; and if negative, the roots are complex. Understanding the discriminant aids in analyzing the factorability of trinomials.
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Related Practice
Textbook Question
Perform the indicated operation and write the answer in decimal notation.
Textbook Question
Find each product. (5−7x)(5+7x)
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Textbook Question
List all numbers from the given set that are a. natural numbers, b. whole numbers, c. integers, d. rational numbers, e. irrational numbers, f. real numbers. {-11, -5/6, 0, 0.75, √5, π, √64}
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Textbook Question
In Exercises 15–58, find each product. (4x2+5x)(4x2−5x)
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Textbook Question
Add or subtract as indicated. (4x−10)/(x−2) − (x−4)/(x−2)
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