Question

1. simplify the rational expression.

\\(\frac{x^2 - 2x - 48}{x^2 - x - 56}\\)
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1. simplify the rational expression.

\\(\frac{-12y^2 + 3y}{9y^2 + 3y}\\)
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Explanation:

Problem 9:

Step1: Factor numerator and denominator

Factor \(x^2 - 2x - 48\): We need two numbers that multiply to \(-48\) and add to \(-2\). Those numbers are \(-8\) and \(6\), so \(x^2 - 2x - 48=(x - 8)(x + 6)\).
Factor \(x^2 - x - 56\): We need two numbers that multiply to \(-56\) and add to \(-1\). Those numbers are \(-8\) and \(7\), so \(x^2 - x - 56=(x - 8)(x + 7)\).
So the expression becomes \(\frac{(x - 8)(x + 6)}{(x - 8)(x + 7)}\).

Step2: Cancel common factors

Cancel the common factor \((x - 8)\) (assuming \(x
eq8\) to avoid division by zero).
The simplified expression is \(\frac{x + 6}{x + 7}\).

Step1: Factor numerator and denominator

Factor numerator \(-12y^2 + 3y\): Factor out \(-3y\), we get \(-3y(4y - 1)\).
Factor denominator \(9y^2 + 3y\): Factor out \(3y\), we get \(3y(3y + 1)\).
So the expression becomes \(\frac{-3y(4y - 1)}{3y(3y + 1)}\).

Step2: Cancel common factors

Cancel the common factor \(3y\) (assuming \(y
eq0\) to avoid division by zero).
The simplified expression is \(\frac{-(4y - 1)}{3y + 1}=\frac{-4y + 1}{3y + 1}\) (or \(\frac{1 - 4y}{3y + 1}\)).

Answer:

\(\frac{x + 6}{x + 7}\)

Problem 10: