Simplify each complex fraction. [ 1/[(x+h)2 + 9] - 1/(x2+9)] / h

Ch. R - Review of Basic Concepts

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 87a

1장, 문제 87a

Simplify each complex fraction. [ 1/[(x+h)² + 9] - 1/(x²+9)] / h

검증된 단계별 안내

Identify the complex fraction: the numerator is the difference of two fractions, \(\frac{1}{(x+h)^2 + 9} - \frac{1}{x^2 + 9}\), and the denominator is \(h\).

Find a common denominator for the two fractions in the numerator, which is \([(x+h)^2 + 9][x^2 + 9]\).

Rewrite the numerator as a single fraction: \(\frac{x^2 + 9 - ((x+h)^2 + 9)}{[(x+h)^2 + 9][x^2 + 9]}\).

Simplify the numerator inside the fraction by expanding and combining like terms: \(x^2 + 9 - (x+h)^2 - 9\).

Rewrite the entire complex fraction as \(\frac{\frac{\text{simplified numerator}}{[(x+h)^2 + 9][x^2 + 9]}}{h}\), then multiply by the reciprocal of \(h\) to get \(\frac{\text{simplified numerator}}{h \cdot [(x+h)^2 + 9][x^2 + 9]}\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Complex Fractions

A complex fraction is a fraction where the numerator, denominator, or both contain fractions themselves. Simplifying involves rewriting the expression as a single fraction by finding common denominators or multiplying numerator and denominator by the least common denominator to eliminate the smaller fractions.

Difference of Fractions

Subtracting fractions requires a common denominator. To subtract 1/[(x+h)^2 + 9] and 1/(x^2 + 9), find the least common denominator, rewrite each fraction with this denominator, then subtract the numerators while keeping the denominator the same.

Simplifying Rational Expressions

After combining fractions, simplify the resulting rational expression by factoring, canceling common factors, and reducing the expression. This step is crucial to express the complex fraction in its simplest form, especially when variables and binomials are involved.