Domanda del libro di testo
In Exercises 111–120, use the order of operations to simplify each expression. 102−100÷52⋅2−3
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Ch. P - Fundamental Concepts of Algebra
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 1, Problema 112
In Exercises 103–114, factor completely. (x+y)4−100(x+y)2
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Recognize that the given expression \((x + y)^4 - 100(x + y)^2\) can be factored by treating \((x + y)^2\) as a single variable. Let \(u = (x + y)^2\), so the expression becomes \(u^2 - 100u\).
Factor out the greatest common factor (GCF) from \(u^2 - 100u\). The GCF is \(u\), so the expression becomes \(u(u - 100)\).
Substitute back \((x + y)^2\) for \(u\). This gives \((x + y)^2((x + y)^2 - 100)\).
Notice that \((x + y)^2 - 100\) is a difference of squares. Use the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), where \(a = (x + y)\) and \(b = 10\). This factors \((x + y)^2 - 100\) into \((x + y - 10)(x + y + 10)\).
Combine all the factors to write the fully factored form of the expression: \((x + y)^2(x + y - 10)(x + y + 10)\).
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Factoring Polynomials
Factoring polynomials involves rewriting a polynomial as a product of simpler polynomials. This process is essential for simplifying expressions and solving equations. In this case, recognizing the structure of the polynomial allows us to apply factoring techniques effectively.
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Difference of Squares
The difference of squares is a specific factoring pattern that states a² - b² = (a - b)(a + b). This concept is crucial for recognizing and simplifying expressions that can be expressed in this form, such as the expression in the given problem, which can be transformed into a difference of squares.
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Substitution Method
The substitution method involves replacing a complex expression with a single variable to simplify the factoring process. In this problem, letting u = (x + y)² can simplify the expression, making it easier to factor and solve. This technique is particularly useful for higher-degree polynomials.
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