Textbook Question
Find each product and write the result in standard form. - 3i(7i - 5)
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 10
Solve each polynomial equation in Exercises 1–10 by factoring and then using the zero-product principle.
Verified step by step guidance1
Start by rewriting the equation to set it equal to zero: \(3x^4 - 81x = 0\).
Factor out the greatest common factor (GCF) from the left side. Identify the GCF of \$3x^4\( and \)81x\(, which is \)3x$, and factor it out: \(3x(x^3 - 27) = 0\).
Recognize that \(x^3 - 27\) is a difference of cubes, since \(27 = 3^3\). Use the difference of cubes formula: \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\), where \(a = x\) and \(b = 3\).
Apply the formula to factor \(x^3 - 27\) as \((x - 3)(x^2 + 3x + 9)\), so the full factorization is \(3x(x - 3)(x^2 + 3x + 9) = 0\).
Use the zero-product principle, which states that if a product of factors equals zero, then at least one of the factors must be zero. Set each factor equal to zero: \(3x = 0\), \(x - 3 = 0\), and \(x^2 + 3x + 9 = 0\), then solve each equation for \(x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Polynomial Equations
A polynomial equation involves expressions with variables raised to whole-number exponents and coefficients. Solving these equations means finding all values of the variable that make the equation true. Understanding the structure of polynomials helps in applying appropriate methods like factoring.
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Factoring Polynomials
Factoring is rewriting a polynomial as a product of simpler polynomials or factors. This process simplifies solving equations by breaking them down into manageable parts. Common factoring techniques include factoring out the greatest common factor, difference of squares, and grouping.
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Zero-Product Principle
The zero-product principle states that if the product of two or more factors equals zero, then at least one of the factors must be zero. This principle allows us to set each factor equal to zero and solve for the variable, providing the solutions to the polynomial equation.
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Related Practice
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Textbook Question
Solve each equation in Exercises 1 - 14 by factoring.
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Textbook Question
Solve each equation in Exercises 1 - 14 by factoring.
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