Textbook Question
Simplify each exponential expression in Exercises 23–64. (−5x^4 y)(−6x^7 y^11)
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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 49
Factor each perfect square trinomial.
Verified step by step guidance1
Recognize that the given expression \(x^2 + 2x + 1\) is a perfect square trinomial. A perfect square trinomial takes the form \(a^2 + 2ab + b^2\) and factors as \((a + b)^2\).
Identify the terms in the trinomial: \(x^2\) is the square of \(x\), and \(1\) is the square of \(1\).
Check the middle term \$2x\( to see if it matches \)2ab\(, where \)a = x\( and \)b = 1$. Since \(2 \times x \times 1 = 2x\), it fits the pattern.
Write the factored form using the pattern \((a + b)^2\), substituting \(a = x\) and \(b = 1\) to get \((x + 1)^2\).
Verify by expanding \((x + 1)^2\) to ensure it equals the original trinomial \(x^2 + 2x + 1\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Perfect Square Trinomial
A perfect square trinomial is a quadratic expression that can be written as the square of a binomial. It takes the form a² + 2ab + b², which factors into (a + b)². Recognizing this pattern helps simplify factoring problems quickly.
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Factoring Quadratic Expressions
Factoring quadratic expressions involves rewriting them as a product of two binomials. For perfect square trinomials, this process is straightforward because the trinomial matches a specific pattern, allowing direct factoring without trial and error.
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Identifying Coefficients and Terms
To factor correctly, it is essential to identify the coefficients of the quadratic and linear terms and the constant term. This helps determine if the trinomial fits the perfect square pattern by checking if the middle term equals twice the product of the square roots of the first and last terms.
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Related Practice
Textbook Question
In Exercises 47 - 49, add or subtract terms whenever possible. 4√72 - 2√48
Textbook Question
Evaluate each expression or indicate that the root is not a real number.
Textbook Question
In Exercises 33–68, add or subtract as indicated.
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Textbook Question
In Exercises 45–54, rationalize the denominator. 13/(3+√11)
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Textbook Question
Determine whether each statement in is true or false. 0 ≥ -6