Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. $\frac{x^3 + x^2}{(x^2 + 4)^2}$

검증된 단계별 안내

Identify the rational expression given: $\frac{x^3 + x^2}{(x^2 + 4)^2}$.

Recognize that the denominator is a repeated irreducible quadratic factor: $(x^2 + 4)^2$.

For a repeated irreducible quadratic factor $(ax^2 + bx + c)^n$, the partial fraction decomposition includes terms of the form $\frac{Bx + C}{ax^2 + bx + c}$, $\frac{Dx + E}{(ax^2 + bx + c)^2}$, and so on, up to the power $n$.

Write the general form of the partial fraction decomposition for this problem as: $\frac{Bx + C}{x^2 + 4} + \frac{Dx + E}{(x^2 + 4)^2}$, where $B$, $C$, $D$, and $E$ are constants to be determined.

Note that since the numerator degree is less than the denominator degree, no polynomial division is necessary before setting up the partial fractions.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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 rational function as a sum of simpler fractions, making integration or other operations easier. It involves breaking down a complex rational expression into a sum of fractions with simpler denominators, typically linear or irreducible quadratic factors.

Repeated Quadratic Factors

When the denominator contains a repeated irreducible quadratic factor, such as (x² + 4)², the decomposition includes terms with the quadratic factor raised to each power up to its multiplicity. For example, terms with denominators (x² + 4) and (x² + 4)² appear, each with linear numerators.

Form of Numerators in Partial Fractions

For linear factors, numerators are constants; for irreducible quadratic factors, numerators are linear expressions (ax + b). This ensures the decomposition can represent any polynomial numerator of lower degree than the denominator factor, allowing for a complete and accurate breakdown.

Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. x3+x2(x2+4)2\(\frac{x^3 + x^2}{(x^2 + 4)^2}\) (x2+4)2x3+x2