Textbook Question
Exercises 177–179 will help you prepare for the material covered in the next section. If is substituted for x in the equation , is the resulting statement true or false?
Ch. 1 - Equations and Inequalities
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 2, Problem 177
Factor completely:
Verified step by step guidance1
Group the terms into two pairs: \( (x^3 + x^2) \) and \( (-4x - 4) \). This is called factoring by grouping.
Factor out the greatest common factor (GCF) from each pair. From \( (x^3 + x^2) \), the GCF is \( x^2 \), and from \( (-4x - 4) \), the GCF is \( -4 \). This gives \( x^2(x + 1) - 4(x + 1) \).
Notice that \( (x + 1) \) is a common factor in both terms. Factor \( (x + 1) \) out, resulting in \( (x + 1)(x^2 - 4) \).
Recognize that \( x^2 - 4 \) is a difference of squares. Use the formula \( a^2 - b^2 = (a - b)(a + b) \) to factor \( x^2 - 4 \) into \( (x - 2)(x + 2) \).
Combine all the factors to write the fully factored form: \( (x + 1)(x - 2)(x + 2) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Factoring Polynomials
Factoring polynomials involves rewriting a polynomial expression as a product of simpler polynomials. This process is essential for simplifying expressions and solving equations. Common methods include factoring by grouping, using the distributive property, and applying special product formulas like the difference of squares.
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Rational Root Theorem
The Rational Root Theorem provides a way to identify possible rational roots of a polynomial equation. It states that any rational solution, expressed as a fraction p/q, must have p as a factor of the constant term and q as a factor of the leading coefficient. This theorem is useful for testing potential roots when factoring.
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Synthetic Division
Synthetic division is a simplified method for dividing polynomials, particularly useful when the divisor is a linear polynomial. It allows for quick calculations to determine if a potential root is indeed a root of the polynomial. If the remainder is zero, the divisor is a factor, facilitating the complete factorization of the polynomial.
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Related Practice
Textbook Question
Solve for x: x^(5/6) + x^(2/3) - 2x^(1/2) = 0
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Textbook Question
Write a quadratic equation in general form whose solution set is {- 3, 5}.
Textbook Question
Exercises 141–143 will help you prepare for the material covered in the next section. If the width of a rectangle is represented by x and the length is represented by x + 200, write a simplified algebraic expression that models the rectangle's perimeter.
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