Simplify each rational expression. (-12 x^2-14 x+10)/(-8...

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Key Concepts

Rational Expressions

A rational expression is a fraction where both the numerator and the denominator are polynomials. Simplifying these expressions involves factoring both the numerator and the denominator and then reducing the fraction by canceling any common factors, making sure to consider restrictions on the variable that would make the denominator zero.

Polynomial Factoring

Factoring polynomials is the process of rewriting a polynomial as a product of its simpler polynomial factors. This is particularly useful when working with quadratic expressions, where methods such as splitting the middle term or using the quadratic formula can reveal factorable components, which are essential for simplification.

Common Factor Cancellation

Once both the numerator and denominator have been factored, any factors that are common to both can be canceled out to simplify the expression. This step must be performed with caution, ensuring that the factors being canceled are not equal to zero, as this maintains the equivalence of the simplified expression to the original.

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