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Express in simplest form:
(2x^(2)-8x-42)/(6x^(2))÷(x^(2)-9)/(x^(2)-3x)

Express in simplest form: $\frac{2 x^{2}-8 x-42}{6 x^{2}} \div \frac{x^{2}-9}{x^{2}-3 x}$

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Divide polynomials using long division

Full solution

Q. Express in simplest form:

\frac{2 x^{2}-8 x-42}{6 x^{2}} \div \frac{x^{2}-9}{x^{2}-3 x}

Multiply by Reciprocal: Simplify the division of fractions by multiplying by the reciprocal. $\newline$ To divide by a fraction, you multiply by its reciprocal. The reciprocal of $(x^{2}-9)/(x^{2}-3x)$ is $(x^{2}-3x)/(x^{2}-9)$ .
Set up Multiplication: Set up the multiplication of the two fractions. $(\frac{2x^{2}-8x-42}{6x^{2}}) \times (\frac{x^{2}-3x}{x^{2}-9})$
Factor Numerators and Denominators: Factor the numerators and denominators where possible. $\newline$ The numerator $2x^{2}-8x-42$ can be factored as $2(x^{2}-4x-21)$ . $\newline$ The denominator $x^{2}-9$ can be factored as $(x+3)(x-3)$ . $\newline$ The denominator $x^{2}-3x$ can be factored as $x(x-3)$ . $\newline$ So the expression becomes: $\newline$ $\frac{2(x^{2}-4x-21)}{6x^{2}} \times \frac{x(x-3)}{(x+3)(x-3)}$
Simplify by Canceling Common Factors: Simplify the expression by canceling out common factors. The $x-3$ in the numerator and denominator cancels out, as does one $x$ from $x(x-3)$ and one $x$ from $6x^{2}$ , leaving a $6$ in the denominator. The expression now looks like this: $\frac{2(x^{2}-4x-21)}{6x} \cdot \frac{x}{x+3}$
Further Simplify Expression: Further simplify the expression. $\newline$ The $2$ in the numerator and the $6$ in the denominator can be simplified to $\frac{1}{3}$ . $\newline$ The expression now looks like this: $\newline$ $\left(\frac{1}{3}\right)\left(x^{2}-4x-21\right)/x \times \frac{x}{x+3}$
Cancel Common X Factor: Cancel out the common $x$ factor in the numerator and denominator. $\newline$ The $x$ in the numerator and the $x$ in the denominator cancel out. $\newline$ The expression now looks like this: $\newline$ $(\frac{1}{3})(x^{2}-4x-21)/(x+3)$
Distribute $\frac{1}{3}$ Across Numerator: Distribute the $\frac{1}{3}$ across the numerator. $\newline$ $\left(\frac{1}{3}\right)x^{2} - \left(\frac{4}{3}\right)x - \frac{7}{(x+3)}$
Check for Further Simplification: Check if the expression can be simplified further. $\newline$ The expression $(\frac{1}{3})x^{2} - (\frac{4}{3})x - \frac{7}{(x+3)}$ is already in its simplest form. There are no common factors that can be canceled out.

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