Textbook Question
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). 4+5(x-7)3, for x =9
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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 8
In Exercises 1–10, factor out the greatest common factor. x(2x+1)+4(2x+1)
Verified step by step guidance1
Step 1: Identify the common factor in the given expression. Notice that both terms, x(2x+1) and 4(2x+1), share the binomial (2x+1) as a common factor.
Step 2: Factor out the greatest common factor, which is (2x+1). This means you rewrite the expression as (2x+1)(x+4).
Step 3: Verify your factorization by distributing (2x+1) back into the terms (x+4). This should return the original expression: x(2x+1) + 4(2x+1).
Step 4: Simplify the factored form if necessary. In this case, the factored form (2x+1)(x+4) is already simplified.
Step 5: Conclude that the expression has been successfully factored, and the greatest common factor has been extracted.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Greatest Common Factor (GCF)
The Greatest Common Factor is the largest expression that divides two or more terms without leaving a remainder. In algebra, identifying the GCF is crucial for simplifying expressions and factoring polynomials. For example, in the expression x(2x+1) + 4(2x+1), the GCF is (2x+1), as it is common to both terms.
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Graphs of Common Functions
Factoring
Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. This technique is essential in algebra for simplifying expressions and solving equations. In the given expression, factoring out the GCF allows us to rewrite it in a more manageable form.
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Distributive Property
The Distributive Property states that a(b + c) = ab + ac, allowing us to distribute a factor across terms within parentheses. This property is fundamental in algebra for expanding expressions and is also used in reverse when factoring. In the expression x(2x+1) + 4(2x+1), recognizing the common factor enables us to apply this property effectively.
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Related Practice
Textbook Question
In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (−6x3+5x2−8x+9)+(17x3+2x2−4x−13)
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Textbook Question
Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √144+25
Textbook Question
In Exercises 5–8, find the degree of the polynomial. x2−4x3+9x−12x4+63
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Textbook Question
Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number. √25−√16
Textbook Question
Evaluate each exponential expression in Exercises 1–22. (−3)0