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Ch. P - Fundamental Concepts of Algebra
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. P - Fundamental Concepts of AlgebraProblema 67
Capitolo 1, Problema 67

In Exercises 33–68, add or subtract as indicated. (4x2+x−6)/(x2+3x+2)−3x/(x+1)+5/(x+2)

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Factorize the denominators where possible. For the first term, \(x^2 + 3x + 2\) can be factored as \((x + 1)(x + 2)\). The other denominators \(x + 1\) and \(x + 2\) are already in their simplest forms.
Identify the least common denominator (LCD). The LCD for the denominators \((x + 1)(x + 2)\), \(x + 1\), and \(x + 2\) is \((x + 1)(x + 2)\).
Rewrite each term with the LCD as the denominator. For the first term, it already has \((x + 1)(x + 2)\) as the denominator. For the second term, multiply both numerator and denominator by \(x + 2\), resulting in \(\frac{3x(x + 2)}{(x + 1)(x + 2)}\). For the third term, multiply both numerator and denominator by \(x + 1\), resulting in \(\frac{5(x + 1)}{(x + 1)(x + 2)}\).
Combine all terms into a single fraction with the common denominator \((x + 1)(x + 2)\). The numerator becomes \((4x^2 + x - 6) - 3x(x + 2) + 5(x + 1)\).
Simplify the numerator by distributing and combining like terms. Expand \(-3x(x + 2)\) and \(5(x + 1)\), then combine all terms in the numerator. The final expression will be a single fraction with the simplified numerator over \((x + 1)(x + 2)\).

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Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. Understanding how to manipulate these expressions, including adding, subtracting, multiplying, and dividing them, is crucial for solving problems involving them. In this question, the rational expressions must be combined, which requires a common denominator.
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Rationalizing Denominators

Finding a Common Denominator

To add or subtract rational expressions, it is essential to find a common denominator. This involves determining the least common multiple (LCM) of the denominators involved. In the given problem, the denominators are (x^2 + 3x + 2), (x + 1), and (x + 2), and finding the LCM will allow for the expressions to be combined correctly.
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Polynomial Factoring

Factoring polynomials is the process of breaking down a polynomial into simpler components, or factors, that can be multiplied together to obtain the original polynomial. In this exercise, factoring the denominator (x^2 + 3x + 2) into (x + 1)(x + 2) is necessary to simplify the rational expressions and facilitate the addition or subtraction.
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Introduction to Factoring Polynomials
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