3. What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following?
c. A factor of (x² + 2x + 6) in the denominator

검증된 단계별 안내

Identify that the factor given, \(x^2 + 2x + 6\), is an irreducible quadratic factor because its discriminant \(\Delta = b^2 - 4ac = 2^2 - 4 \cdot 1 \cdot 6 = 4 - 24 = -20 < 0\), which means it cannot be factored further over the real numbers.

Recall that for an irreducible quadratic factor \(ax^2 + bx + c\) in the denominator, the corresponding term in the partial fraction decomposition has the form \(\frac{Ax + B}{ax^2 + bx + c}\), where \(A\) and \(B\) are constants to be determined.

If the factor \(x^2 + 2x + 6\) appears with multiplicity 1 (i.e., it is not repeated), then the partial fraction term is simply \(\frac{Ax + B}{x^2 + 2x + 6}\).

If the factor appears with multiplicity greater than 1, say \(n\), then the decomposition includes terms for each power from 1 up to \(n\): \(\frac{A_1x + B_1}{x^2 + 2x + 6} + \frac{A_2x + B_2}{(x^2 + 2x + 6)^2} + \cdots + \frac{A_nx + B_n}{(x^2 + 2x + 6)^n}\).

Summarize that the key point is that each irreducible quadratic factor in the denominator corresponds to a numerator that is a linear polynomial \(Ax + B\) over that quadratic factor in the partial fraction decomposition.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Partial Fraction Decomposition

Partial fraction decomposition is a method used to express a proper rational function as a sum of simpler fractions whose denominators are factors of the original denominator. This technique simplifies integration and other operations by breaking complex fractions into manageable parts.

Irreducible Quadratic Factors

An irreducible quadratic factor is a quadratic polynomial that cannot be factored further over the real numbers. In partial fraction decomposition, such factors in the denominator correspond to terms with linear numerators, for example, (Ax + B)/(quadratic factor).

Proper Rational Function

A proper rational function is a fraction where the degree of the numerator is less than the degree of the denominator. This condition ensures that partial fraction decomposition can be applied directly without polynomial division.

3. What term(s) should appear in the partial fraction decomposition of a proper rational function with each of the following?c. A factor of (x² + 2x + 6) in the denominator