Add or subtract, as indicated. See Example 4. 4/(x+1) + 1/(x² - x + 1) - 12/(x³ + 1)

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Identify the three rational expressions to be combined: \(\frac{4}{x+1}\), \(\frac{1}{x^{2} - x + 1}\), and \(\frac{12}{x^{3} + 1}\).

Factor the denominator \(x^{3} + 1\) using the sum of cubes formula: \(a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})\). Here, \(x^{3} + 1 = (x + 1)(x^{2} - x + 1)\).

Recognize that the denominators \(x + 1\) and \(x^{2} - x + 1\) are factors of \(x^{3} + 1\), so the least common denominator (LCD) for all three fractions is \(x^{3} + 1 = (x + 1)(x^{2} - x + 1)\).

Rewrite each fraction with the LCD as the denominator by multiplying numerator and denominator appropriately: - For \(\frac{4}{x+1}\), multiply numerator and denominator by \(x^{2} - x + 1\). - For \(\frac{1}{x^{2} - x + 1}\), multiply numerator and denominator by \(x + 1\). - The third fraction already has the LCD as denominator.

Combine the numerators over the common denominator \(x^{3} + 1\) by performing the indicated addition and subtraction, then simplify the resulting numerator if possible.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Polynomial Addition and Subtraction

This concept involves combining like terms of polynomials by adding or subtracting their coefficients. When dealing with rational expressions, the numerators are combined after ensuring a common denominator. Understanding how to manipulate polynomial expressions is essential for simplifying the given expression.

Finding a Common Denominator

To add or subtract rational expressions, their denominators must be the same. This requires finding the least common denominator (LCD), often by factoring each denominator and taking the product of unique factors at their highest powers. This step allows the expressions to be combined into a single rational expression.

Factoring Polynomials

Factoring breaks down polynomials into products of simpler polynomials or expressions. Recognizing common factors or special products (like difference of squares or sum/difference of cubes) helps in simplifying denominators and finding the LCD. Factoring is crucial for simplifying the final expression after addition or subtraction.