Textbook Question
Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
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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 12
In Exercises 11–16, factor by grouping. x3−3x2+4x−12
Verified step by step guidance1
Group the terms into two pairs: \( (x^3 - 3x^2) + (4x - 12) \).
Factor out the greatest common factor (GCF) from each group: \( x^2(x - 3) + 4(x - 3) \).
Notice that \( (x - 3) \) is a common factor in both terms. Factor \( (x - 3) \) out: \( (x - 3)(x^2 + 4) \).
Verify the factorization by distributing \( (x - 3) \) back into \( (x^2 + 4) \) to ensure it equals the original expression.
The expression is now factored as \( (x - 3)(x^2 + 4) \). This is the final factored form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring by Grouping
Factoring by grouping is a method used to factor polynomials by rearranging and grouping terms in pairs. This technique involves identifying common factors in each group, allowing for the extraction of a common binomial factor. It is particularly useful for polynomials with four or more terms, as it simplifies the expression into a product of simpler polynomials.
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04:36
Factor by Grouping
Common Factors
A common factor is a number or variable that divides two or more terms without leaving a remainder. In the context of polynomials, identifying common factors within grouped terms is essential for simplifying the expression. Recognizing these factors allows for the extraction of a binomial or monomial, which is a crucial step in the factoring process.
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Polynomial Expressions
Polynomial expressions are mathematical expressions that consist of variables raised to non-negative integer powers, combined using addition, subtraction, and multiplication. Understanding the structure of polynomials, including their degree and coefficients, is vital for applying factoring techniques. In the given expression, recognizing the degree and the arrangement of terms aids in effectively applying the grouping method.
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Related Practice
Textbook Question
State the name of the property illustrated: 3+17 = 17+3
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Textbook Question
In Exercises 9–14, perform the indicated operations. Write the resulting polynomial in standard form and indicate its degree. (5x2−7x−8)+(2x2−3x+7)−(x2−4x−3)
Textbook Question
Evaluate each expression in Exercises 1–12, or indicate that the root is not a real number.
Textbook Question
In Exercises 1–16, evaluate each algebraic expression for the given value or values of the variable(s). 5(x+2)/(2x-14), for x=10
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Textbook Question
Express the distance between the numbers -17 and 4 using absolute value. Then evaluate the absolute value.