Simplify each complex fraction. 3p2−16+p1p−4\frac{\frac{3}{$$p^2 - 16$$} + p}{\frac{1}{p - 4}} p−41p2−163+p

Ch. R - Review of Basic Concepts

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Lial 13th Edition

Ch. R - Review of Basic Concepts

문제 81

1장, 문제 81

Simplify each complex fraction. $$\frac{\frac{3}{p^2 - 16}$ + p}{$\frac{1}{p - 4}$}$

검증된 단계별 안내

Identify the complex fraction: $\frac{\frac{3}{p^{2} - 16} + p}{\frac{1}{p - 4}}$.

Recognize that $p^{2} - 16$ is a difference of squares and factor it as $p^{2} - 16 = (p - 4)(p + 4)$.

Rewrite the numerator by expressing $p$ as a fraction with denominator $(p - 4)(p + 4)$ to combine the terms: $\frac{3}{(p - 4)(p + 4)} + \frac{p(p - 4)(p + 4)}{(p - 4)(p + 4)}$.

Combine the fractions in the numerator over the common denominator $(p - 4)(p + 4)$: $\frac{3 + p(p - 4)(p + 4)}{(p - 4)(p + 4)}$.

Divide the combined numerator by the denominator $\frac{1}{p - 4}$ by multiplying the numerator by the reciprocal of the denominator: $\frac{3 + p(p - 4)(p + 4)}{(p - 4)(p + 4)} \times (p - 4)$, then simplify the expression.

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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 to eliminate the smaller fractions, often by finding a common denominator or multiplying numerator and denominator by the least common denominator (LCD).

Factoring Polynomials

Factoring involves expressing a polynomial as a product of simpler polynomials. Recognizing special forms like the difference of squares, e.g., p^2 - 16 = (p - 4)(p + 4), helps simplify expressions and cancel common factors in fractions.

Operations with Rational Expressions

Rational expressions are fractions with polynomials in numerator and denominator. Adding, subtracting, multiplying, or dividing them requires finding common denominators, factoring, and simplifying by canceling common factors to reduce the expression to simplest form.

Simplify each complex fraction. 3p2−16+p1p−4\(\frac{\frac{3}{p^2 - 16}\) + p}{\(\frac{1}{p - 4}\)} p−41p2−163+p

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Complex Fractions

Factoring Polynomials

Operations with Rational Expressions

Simplify each complex fraction. $\(\frac{\frac{3}{p^2 - 16}\) + p}{\(\frac{1}{p - 4}\)}$