QuestionAugust 27, 2025

Express the quotient in simplest form. (x^2-4)/(x^3)+7x^(2)div (x^3-x^2-6x)/(x^{2)+4 A. ((x-2)(x-3))/(x^3)(x+3) B. (x+2)/(x^3) C. (x-2)/(x^3) D. (x-2)/(x^2)

Express the quotient in simplest form. (x^2-4)/(x^3)+7x^(2)div (x^3-x^2-6x)/(x^{2)+4 A. ((x-2)(x-3))/(x^3)(x+3) B. (x+2)/(x^3) C. (x-2)/(x^3) D. (x-2)/(x^2)
Express the quotient in simplest form. (x^2-4)/(x^3)+7x^(2)div (x^3-x^2-6x)/(x^{2)+4 A. ((x-2)(x-3))/(x^3)(x+3) B. (x+2)/(x^3) C. (x-2)/(x^3) D. (x-2)/(x^2)

Solution
4.6(318 votes)

Answer

None of the given options match the simplified expression. Explanation 1. Simplify the division of fractions Use the rule \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} to rewrite the expression as \frac{x^2 - 4}{x^3 + 7x^2} \times \frac{x^2 + 4}{x^3 - x^2 - 6x}. 2. Factor numerators and denominators Factor x^2 - 4 as (x-2)(x+2), x^3 + 7x^2 as x^2(x+7), x^3 - x^2 - 6x as x(x-3)(x+2). 3. Substitute factored forms into the expression The expression becomes \frac{(x-2)(x+2)}{x^2(x+7)} \times \frac{x^2 + 4}{x(x-3)(x+2)}. 4. Cancel common factors Cancel (x+2) from numerator and denominator. The expression simplifies to \frac{(x-2)(x^2 + 4)}{x^3(x-3)}. 5. Simplify further if possible No further simplification is possible. The simplest form is \frac{(x-2)(x^2 + 4)}{x^3(x-3)}.

Explanation

1. Simplify the division of fractions<br /> Use the rule $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$ to rewrite the expression as $\frac{x^2 - 4}{x^3 + 7x^2} \times \frac{x^2 + 4}{x^3 - x^2 - 6x}$.<br /><br />2. Factor numerators and denominators<br /> Factor $x^2 - 4$ as $(x-2)(x+2)$, $x^3 + 7x^2$ as $x^2(x+7)$, $x^3 - x^2 - 6x$ as $x(x-3)(x+2)$.<br /><br />3. Substitute factored forms into the expression<br /> The expression becomes $\frac{(x-2)(x+2)}{x^2(x+7)} \times \frac{x^2 + 4}{x(x-3)(x+2)}$.<br /><br />4. Cancel common factors<br /> Cancel $(x+2)$ from numerator and denominator. The expression simplifies to $\frac{(x-2)(x^2 + 4)}{x^3(x-3)}$.<br /><br />5. Simplify further if possible<br /> No further simplification is possible. The simplest form is $\frac{(x-2)(x^2 + 4)}{x^3(x-3)}$.
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